Edited from The De Morgan Gazette no () ndashbitly2hlyHzZ ISSN ndash

Thales and the

Nine-point Conic

David Pierce

December

July

March

Mathematics Department

Mimar Sinan Fine Arts University

httpmatmsgsuedutr~dpierce

Abstract

The nine-point circle is established by Euclidean means thenine-point conic Cartesian Cartesian geometry is developedfrom Euclidean by means of Thalesrsquos Theorem A theory ofproportion is given and Thalesrsquos Theorem proved on the ba-sis of Book i of Euclidrsquos Elements without the Archimedeanassumption of Book v Euclidrsquos theory of areas is used al-though this is obviated by Hilbertrsquos theory of lengths It isobserved how Apollonius relies on Euclidrsquos theory of areasThe historical foundations of the name of Thalesrsquos Theoremare considered Thales is thought to have identified water as auniversal substrate his recognition of mathematical theoremsas such represents a similar unification of things

Preface

I review here the theorem dating to the early nineteenth cen-tury that a certain collection of nine points associated with anarbitrary triangle lie on a circle Some of the nine points aredefined by means of the orthocenter of the triangle If the or-thocenter is replaced by an arbitrary point not collinear withany two vertices of the triangle then the same nine pointsstill lie on a conic section This was recognized in the latenineteenth century and I review this result too All of this isan occasion to consider the historical and logical developmentof geometry as seen in work of Thales Euclid ApolloniusDescartes Hilbert and Hartshorne All of the proofs hereare based ultimately and usually explicitly on Book i of Eu-clidrsquos Elements Even Descartesrsquos ldquoanalyticrdquo methods are sojustified although Descartes implicitly used the theory of pro-portion found in Euclidrsquos Book v In particular Descartesrelied on what is in some countries today called Thalesrsquos The-orem Using only Book i of Euclid and in particular its theoryof areas I give a proof of Thalesrsquos Theorem in the followingform

If two triangles share a common angle the bases of the tri-angles are parallel if and only if the rectangles bounded byalternate sides of the common angle are equal

This allows an alternative development of a theory of propor-tion

An even greater logical simplification is possible As Hilbertshows and as I review using the simpler arguments ofHartshorne one can actually prove Euclidrsquos assumption thata part of a bounded planar region is less than the whole re-gion One still has to assume that a part of a bounded straightline is less than the whole but one can now conceive of thetheory of proportion as based entirely on the notion of lengthThis accords with Descartesrsquos geometry where length is thefundamental notion

For Apollonius by contrast as I show with an exampleareas are essential His proofs about conic sections are basedon areas In Cartesian geometry we assign letters to lengthsknown and unknown so as to be able to do computationswithout any visualization This is how we obtain the nine-point conic

We may however consider what is lost when we do not di-rectly visually consider areas Several theorems attributed toThales seem to be due to a visual appreciation of symmetryI end with this topic one that I consider further in [] anarticle not yet published when the present essay was published

Aside from the formatting (I prefer A paper so that twopages can easily be read side by side on a computer screen orfor that matter on an A printout)mdashaside from the format-ting the present version of my essay differs from the versionpublished in the De Morgan Gazette as follows

Reference [] is now to a published paper not to thepreliminary version on my own webpage

In the Introduction to the last clause of the rd para-graph I have supplied the missing ldquonotrdquo so that theclause now reads ldquodoing this does not represent the ac-complishment of a preconceived goalrdquo

In the last paragraph of sect to what was a bare refer-

ence to Elements i I have added notice of Proclusrsquosattribution to Thales and a reference to Chapter

In Figure in sect labelled points of intersection areno longer emphasized with dots

In secta) Towards the end of the first paragraph where the

relation defined by () is observed to be transitiveif we allow passage to a third dimension the originalpassage was to a fourth dimension in which a middot d middote middot f = b middot c middot d middot e

b) in the paragraph before Theorem after ldquoThe def-inition also ensures that () implies ()rdquo I haveadded the clause ldquolet (x y) = (c d) in ()rdquo

In secta) the displayed proportion (a b) amp (b c) a c is

no longer numberedb) I have deleted the second ldquothisrdquo in the garbled sen-

tence ldquoDescartes shows this implicitly this in orderto solve ancient problemsrdquo

In secta) at the end of the first paragraph the reference to

ldquop rdquo now has the proper spacing (the TEX codeis now p284 but I had forgotten the backslashbefore)

b) in the second paragraph in the clause that was ldquothepoint now designed by (x y) is the one that wascalled (x minus hya y) beforerdquo I have corrected ldquode-signedrdquo to ldquodesignatedrdquo and made the minus sign aplus sign

In secta) I have enlarged and expanded Figure b) in the fourth paragraph the clause beginning ldquoIf d

is the distance between the new originrdquo continuesnow with ldquoandrdquo (not ldquotordquo)

c) in the fifth paragraph the letter Β at the end is nowthus a Beta rather than B

d) in the sixth paragraph after the beginning ldquoA lengthΗrdquo the parenthesis ldquonot depictedrdquo is added

e) the proportion () formerly had the same internallabel as () which meant that the ensuing proof

of () was wrongly justified by a reference to itselfrather than to ()

In the third paragraph of Chapter ldquodirectionrdquo has thusbeen made singular

Contents

Introduction

The Nine-point Circle

Centers of a triangle The angle in the semicircle The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem Thales and Desargues Locus problems The nine-point conic

Lengths and Areas

Algebra Apollonius

Unity

Bibliography

List of Figures

The nine points A nine-point hyperbola A nine-point ellipse

The arbelos Concurrence of altitudes Bisector of two sides of a triangle Concurrence of medians The angle in the semicircle Right angles and circles The nine-point circle

Elements i The rectangular case of Thalesrsquos Theorem Intermediate form of Thalesrsquos Theorem Converse of Elements i Thalesrsquos Theorem Transitivity Desarguesrsquos Theorem The quadratrix Solution of a five-line locus problem

Hilbertrsquos multiplication Pappusrsquos Theorem Hilbertrsquos trigonometry

Associativity and commutativity Distributivity Circle inscribed in triangle Area of triangle in two ways Change of coordinates Proposition i of Apollonius Proposition i of Apollonius

Symmetries

List of Figures

Introduction

According to Herodotus of Halicarnassus a war was ended bya solar eclipse and Thales of Miletus had predicted the year[ i] The war was between the Lydians and the Medes inAsia Minor the year was the sixth of the war The eclipse isthought to be the one that must have occurred on May ofthe Julian calendar in the year before the Common Era[ p n ]

The birth of Thales is sometimes assigned to the year bce This is done in the ldquoThalesrdquo article on Wikipedia []but it was also done in ancient times (according to the reck-oning of years by Olympiads) The sole reason for this assign-ment seems to be the assumption that Thales must have beenforty when he predicted the eclipse [ p n ]

The nine-point circle was found early in the nineteenth cen-tury of the Common Era The -point conic a generalizationwas found later in the century The existence of the curvesis proved with mathematics that can be traced to Thales andthat is learned today in high school

In the Euclidean plane every triangle determines a fewpoints individually the incenter circumcenter orthocenterand centroid In addition to the vertices themselves the tri-angle also determines various triples of points such as thosein Figure the feet bull of the altitudes the midpoints ofthe sides and the midpoints N of the straight lines runningfrom the orthocenter to the three vertices It turns out that

b

b

b

l

ll

u

u u

Figure The nine points

these nine points lie on a circle called the nine-point circle

or Euler circle of the triangle The discovery of this circleseems to have been published first by Brianchon and Ponceletin ndash [] then by Feuerbach in [ ]

Precise references for the discovery of the nine-point circleare given in Boyerrsquos History of Mathematics [ pp ndash]and Klinersquos Mathematical Thought from Ancient to Modern

Times [ p ] However as with the birth of Thalesprecision is different from accuracy Kline attributes the firstpublication on the nine-point circle to ldquoGergonne and Pon-celetrdquo In consulting his notes Kline may have confused anauthor with the publisher who was himself a mathematicianBoyer mentions that the joint paper of Brianchon and Ponceletwas published in Gergonnersquos Annales The confusion here is areminder that even seemingly authoritative sources may be inerror

The mathematician is supposed to be a skeptic accepting

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Abstract

The nine-point circle is established by Euclidean means thenine-point conic Cartesian Cartesian geometry is developedfrom Euclidean by means of Thalesrsquos Theorem A theory ofproportion is given and Thalesrsquos Theorem proved on the ba-sis of Book i of Euclidrsquos Elements without the Archimedeanassumption of Book v Euclidrsquos theory of areas is used al-though this is obviated by Hilbertrsquos theory of lengths It isobserved how Apollonius relies on Euclidrsquos theory of areasThe historical foundations of the name of Thalesrsquos Theoremare considered Thales is thought to have identified water as auniversal substrate his recognition of mathematical theoremsas such represents a similar unification of things

Preface

I review here the theorem dating to the early nineteenth cen-tury that a certain collection of nine points associated with anarbitrary triangle lie on a circle Some of the nine points aredefined by means of the orthocenter of the triangle If the or-thocenter is replaced by an arbitrary point not collinear withany two vertices of the triangle then the same nine pointsstill lie on a conic section This was recognized in the latenineteenth century and I review this result too All of this isan occasion to consider the historical and logical developmentof geometry as seen in work of Thales Euclid ApolloniusDescartes Hilbert and Hartshorne All of the proofs hereare based ultimately and usually explicitly on Book i of Eu-clidrsquos Elements Even Descartesrsquos ldquoanalyticrdquo methods are sojustified although Descartes implicitly used the theory of pro-portion found in Euclidrsquos Book v In particular Descartesrelied on what is in some countries today called Thalesrsquos The-orem Using only Book i of Euclid and in particular its theoryof areas I give a proof of Thalesrsquos Theorem in the followingform

If two triangles share a common angle the bases of the tri-angles are parallel if and only if the rectangles bounded byalternate sides of the common angle are equal

This allows an alternative development of a theory of propor-tion

An even greater logical simplification is possible As Hilbertshows and as I review using the simpler arguments ofHartshorne one can actually prove Euclidrsquos assumption thata part of a bounded planar region is less than the whole re-gion One still has to assume that a part of a bounded straightline is less than the whole but one can now conceive of thetheory of proportion as based entirely on the notion of lengthThis accords with Descartesrsquos geometry where length is thefundamental notion

For Apollonius by contrast as I show with an exampleareas are essential His proofs about conic sections are basedon areas In Cartesian geometry we assign letters to lengthsknown and unknown so as to be able to do computationswithout any visualization This is how we obtain the nine-point conic

We may however consider what is lost when we do not di-rectly visually consider areas Several theorems attributed toThales seem to be due to a visual appreciation of symmetryI end with this topic one that I consider further in [] anarticle not yet published when the present essay was published

Aside from the formatting (I prefer A paper so that twopages can easily be read side by side on a computer screen orfor that matter on an A printout)mdashaside from the format-ting the present version of my essay differs from the versionpublished in the De Morgan Gazette as follows

Reference [] is now to a published paper not to thepreliminary version on my own webpage

In the Introduction to the last clause of the rd para-graph I have supplied the missing ldquonotrdquo so that theclause now reads ldquodoing this does not represent the ac-complishment of a preconceived goalrdquo

In the last paragraph of sect to what was a bare refer-

ence to Elements i I have added notice of Proclusrsquosattribution to Thales and a reference to Chapter

In Figure in sect labelled points of intersection areno longer emphasized with dots

In secta) Towards the end of the first paragraph where the

relation defined by () is observed to be transitiveif we allow passage to a third dimension the originalpassage was to a fourth dimension in which a middot d middote middot f = b middot c middot d middot e

b) in the paragraph before Theorem after ldquoThe def-inition also ensures that () implies ()rdquo I haveadded the clause ldquolet (x y) = (c d) in ()rdquo

In secta) the displayed proportion (a b) amp (b c) a c is

no longer numberedb) I have deleted the second ldquothisrdquo in the garbled sen-

tence ldquoDescartes shows this implicitly this in orderto solve ancient problemsrdquo

In secta) at the end of the first paragraph the reference to

ldquop rdquo now has the proper spacing (the TEX codeis now p284 but I had forgotten the backslashbefore)

b) in the second paragraph in the clause that was ldquothepoint now designed by (x y) is the one that wascalled (x minus hya y) beforerdquo I have corrected ldquode-signedrdquo to ldquodesignatedrdquo and made the minus sign aplus sign

In secta) I have enlarged and expanded Figure b) in the fourth paragraph the clause beginning ldquoIf d

is the distance between the new originrdquo continuesnow with ldquoandrdquo (not ldquotordquo)

c) in the fifth paragraph the letter Β at the end is nowthus a Beta rather than B

d) in the sixth paragraph after the beginning ldquoA lengthΗrdquo the parenthesis ldquonot depictedrdquo is added

e) the proportion () formerly had the same internallabel as () which meant that the ensuing proof

of () was wrongly justified by a reference to itselfrather than to ()

In the third paragraph of Chapter ldquodirectionrdquo has thusbeen made singular

Contents

Introduction

The Nine-point Circle

Centers of a triangle The angle in the semicircle The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem Thales and Desargues Locus problems The nine-point conic

Lengths and Areas

Algebra Apollonius

Unity

Bibliography

List of Figures

The nine points A nine-point hyperbola A nine-point ellipse

The arbelos Concurrence of altitudes Bisector of two sides of a triangle Concurrence of medians The angle in the semicircle Right angles and circles The nine-point circle

Elements i The rectangular case of Thalesrsquos Theorem Intermediate form of Thalesrsquos Theorem Converse of Elements i Thalesrsquos Theorem Transitivity Desarguesrsquos Theorem The quadratrix Solution of a five-line locus problem

Hilbertrsquos multiplication Pappusrsquos Theorem Hilbertrsquos trigonometry

Associativity and commutativity Distributivity Circle inscribed in triangle Area of triangle in two ways Change of coordinates Proposition i of Apollonius Proposition i of Apollonius

Symmetries

List of Figures

Introduction

According to Herodotus of Halicarnassus a war was ended bya solar eclipse and Thales of Miletus had predicted the year[ i] The war was between the Lydians and the Medes inAsia Minor the year was the sixth of the war The eclipse isthought to be the one that must have occurred on May ofthe Julian calendar in the year before the Common Era[ p n ]

The birth of Thales is sometimes assigned to the year bce This is done in the ldquoThalesrdquo article on Wikipedia []but it was also done in ancient times (according to the reck-oning of years by Olympiads) The sole reason for this assign-ment seems to be the assumption that Thales must have beenforty when he predicted the eclipse [ p n ]

The nine-point circle was found early in the nineteenth cen-tury of the Common Era The -point conic a generalizationwas found later in the century The existence of the curvesis proved with mathematics that can be traced to Thales andthat is learned today in high school

In the Euclidean plane every triangle determines a fewpoints individually the incenter circumcenter orthocenterand centroid In addition to the vertices themselves the tri-angle also determines various triples of points such as thosein Figure the feet bull of the altitudes the midpoints ofthe sides and the midpoints N of the straight lines runningfrom the orthocenter to the three vertices It turns out that

b

b

b

l

ll

u

u u

Figure The nine points

these nine points lie on a circle called the nine-point circle

or Euler circle of the triangle The discovery of this circleseems to have been published first by Brianchon and Ponceletin ndash [] then by Feuerbach in [ ]

Precise references for the discovery of the nine-point circleare given in Boyerrsquos History of Mathematics [ pp ndash]and Klinersquos Mathematical Thought from Ancient to Modern

Times [ p ] However as with the birth of Thalesprecision is different from accuracy Kline attributes the firstpublication on the nine-point circle to ldquoGergonne and Pon-celetrdquo In consulting his notes Kline may have confused anauthor with the publisher who was himself a mathematicianBoyer mentions that the joint paper of Brianchon and Ponceletwas published in Gergonnersquos Annales The confusion here is areminder that even seemingly authoritative sources may be inerror

The mathematician is supposed to be a skeptic accepting

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Preface

I review here the theorem dating to the early nineteenth cen-tury that a certain collection of nine points associated with anarbitrary triangle lie on a circle Some of the nine points aredefined by means of the orthocenter of the triangle If the or-thocenter is replaced by an arbitrary point not collinear withany two vertices of the triangle then the same nine pointsstill lie on a conic section This was recognized in the latenineteenth century and I review this result too All of this isan occasion to consider the historical and logical developmentof geometry as seen in work of Thales Euclid ApolloniusDescartes Hilbert and Hartshorne All of the proofs hereare based ultimately and usually explicitly on Book i of Eu-clidrsquos Elements Even Descartesrsquos ldquoanalyticrdquo methods are sojustified although Descartes implicitly used the theory of pro-portion found in Euclidrsquos Book v In particular Descartesrelied on what is in some countries today called Thalesrsquos The-orem Using only Book i of Euclid and in particular its theoryof areas I give a proof of Thalesrsquos Theorem in the followingform

If two triangles share a common angle the bases of the tri-angles are parallel if and only if the rectangles bounded byalternate sides of the common angle are equal

This allows an alternative development of a theory of propor-tion

An even greater logical simplification is possible As Hilbertshows and as I review using the simpler arguments ofHartshorne one can actually prove Euclidrsquos assumption thata part of a bounded planar region is less than the whole re-gion One still has to assume that a part of a bounded straightline is less than the whole but one can now conceive of thetheory of proportion as based entirely on the notion of lengthThis accords with Descartesrsquos geometry where length is thefundamental notion

For Apollonius by contrast as I show with an exampleareas are essential His proofs about conic sections are basedon areas In Cartesian geometry we assign letters to lengthsknown and unknown so as to be able to do computationswithout any visualization This is how we obtain the nine-point conic

We may however consider what is lost when we do not di-rectly visually consider areas Several theorems attributed toThales seem to be due to a visual appreciation of symmetryI end with this topic one that I consider further in [] anarticle not yet published when the present essay was published

Aside from the formatting (I prefer A paper so that twopages can easily be read side by side on a computer screen orfor that matter on an A printout)mdashaside from the format-ting the present version of my essay differs from the versionpublished in the De Morgan Gazette as follows

Reference [] is now to a published paper not to thepreliminary version on my own webpage

In the Introduction to the last clause of the rd para-graph I have supplied the missing ldquonotrdquo so that theclause now reads ldquodoing this does not represent the ac-complishment of a preconceived goalrdquo

In the last paragraph of sect to what was a bare refer-

ence to Elements i I have added notice of Proclusrsquosattribution to Thales and a reference to Chapter

In Figure in sect labelled points of intersection areno longer emphasized with dots

In secta) Towards the end of the first paragraph where the

relation defined by () is observed to be transitiveif we allow passage to a third dimension the originalpassage was to a fourth dimension in which a middot d middote middot f = b middot c middot d middot e

b) in the paragraph before Theorem after ldquoThe def-inition also ensures that () implies ()rdquo I haveadded the clause ldquolet (x y) = (c d) in ()rdquo

In secta) the displayed proportion (a b) amp (b c) a c is

no longer numberedb) I have deleted the second ldquothisrdquo in the garbled sen-

tence ldquoDescartes shows this implicitly this in orderto solve ancient problemsrdquo

In secta) at the end of the first paragraph the reference to

ldquop rdquo now has the proper spacing (the TEX codeis now p284 but I had forgotten the backslashbefore)

b) in the second paragraph in the clause that was ldquothepoint now designed by (x y) is the one that wascalled (x minus hya y) beforerdquo I have corrected ldquode-signedrdquo to ldquodesignatedrdquo and made the minus sign aplus sign

In secta) I have enlarged and expanded Figure b) in the fourth paragraph the clause beginning ldquoIf d

is the distance between the new originrdquo continuesnow with ldquoandrdquo (not ldquotordquo)

c) in the fifth paragraph the letter Β at the end is nowthus a Beta rather than B

d) in the sixth paragraph after the beginning ldquoA lengthΗrdquo the parenthesis ldquonot depictedrdquo is added

e) the proportion () formerly had the same internallabel as () which meant that the ensuing proof

of () was wrongly justified by a reference to itselfrather than to ()

In the third paragraph of Chapter ldquodirectionrdquo has thusbeen made singular

Contents

Introduction

The Nine-point Circle

Centers of a triangle The angle in the semicircle The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem Thales and Desargues Locus problems The nine-point conic

Lengths and Areas

Algebra Apollonius

Unity

Bibliography

List of Figures

The nine points A nine-point hyperbola A nine-point ellipse

The arbelos Concurrence of altitudes Bisector of two sides of a triangle Concurrence of medians The angle in the semicircle Right angles and circles The nine-point circle

Elements i The rectangular case of Thalesrsquos Theorem Intermediate form of Thalesrsquos Theorem Converse of Elements i Thalesrsquos Theorem Transitivity Desarguesrsquos Theorem The quadratrix Solution of a five-line locus problem

Hilbertrsquos multiplication Pappusrsquos Theorem Hilbertrsquos trigonometry

Associativity and commutativity Distributivity Circle inscribed in triangle Area of triangle in two ways Change of coordinates Proposition i of Apollonius Proposition i of Apollonius

Symmetries

List of Figures

Introduction

According to Herodotus of Halicarnassus a war was ended bya solar eclipse and Thales of Miletus had predicted the year[ i] The war was between the Lydians and the Medes inAsia Minor the year was the sixth of the war The eclipse isthought to be the one that must have occurred on May ofthe Julian calendar in the year before the Common Era[ p n ]

The birth of Thales is sometimes assigned to the year bce This is done in the ldquoThalesrdquo article on Wikipedia []but it was also done in ancient times (according to the reck-oning of years by Olympiads) The sole reason for this assign-ment seems to be the assumption that Thales must have beenforty when he predicted the eclipse [ p n ]

The nine-point circle was found early in the nineteenth cen-tury of the Common Era The -point conic a generalizationwas found later in the century The existence of the curvesis proved with mathematics that can be traced to Thales andthat is learned today in high school

In the Euclidean plane every triangle determines a fewpoints individually the incenter circumcenter orthocenterand centroid In addition to the vertices themselves the tri-angle also determines various triples of points such as thosein Figure the feet bull of the altitudes the midpoints ofthe sides and the midpoints N of the straight lines runningfrom the orthocenter to the three vertices It turns out that

b

b

b

l

ll

u

u u

Figure The nine points

these nine points lie on a circle called the nine-point circle

or Euler circle of the triangle The discovery of this circleseems to have been published first by Brianchon and Ponceletin ndash [] then by Feuerbach in [ ]

Precise references for the discovery of the nine-point circleare given in Boyerrsquos History of Mathematics [ pp ndash]and Klinersquos Mathematical Thought from Ancient to Modern

Times [ p ] However as with the birth of Thalesprecision is different from accuracy Kline attributes the firstpublication on the nine-point circle to ldquoGergonne and Pon-celetrdquo In consulting his notes Kline may have confused anauthor with the publisher who was himself a mathematicianBoyer mentions that the joint paper of Brianchon and Ponceletwas published in Gergonnersquos Annales The confusion here is areminder that even seemingly authoritative sources may be inerror

The mathematician is supposed to be a skeptic accepting

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

An even greater logical simplification is possible As Hilbertshows and as I review using the simpler arguments ofHartshorne one can actually prove Euclidrsquos assumption thata part of a bounded planar region is less than the whole re-gion One still has to assume that a part of a bounded straightline is less than the whole but one can now conceive of thetheory of proportion as based entirely on the notion of lengthThis accords with Descartesrsquos geometry where length is thefundamental notion

For Apollonius by contrast as I show with an exampleareas are essential His proofs about conic sections are basedon areas In Cartesian geometry we assign letters to lengthsknown and unknown so as to be able to do computationswithout any visualization This is how we obtain the nine-point conic

We may however consider what is lost when we do not di-rectly visually consider areas Several theorems attributed toThales seem to be due to a visual appreciation of symmetryI end with this topic one that I consider further in [] anarticle not yet published when the present essay was published

Aside from the formatting (I prefer A paper so that twopages can easily be read side by side on a computer screen orfor that matter on an A printout)mdashaside from the format-ting the present version of my essay differs from the versionpublished in the De Morgan Gazette as follows

Reference [] is now to a published paper not to thepreliminary version on my own webpage

In the Introduction to the last clause of the rd para-graph I have supplied the missing ldquonotrdquo so that theclause now reads ldquodoing this does not represent the ac-complishment of a preconceived goalrdquo

In the last paragraph of sect to what was a bare refer-

ence to Elements i I have added notice of Proclusrsquosattribution to Thales and a reference to Chapter

In Figure in sect labelled points of intersection areno longer emphasized with dots

In secta) Towards the end of the first paragraph where the

relation defined by () is observed to be transitiveif we allow passage to a third dimension the originalpassage was to a fourth dimension in which a middot d middote middot f = b middot c middot d middot e

b) in the paragraph before Theorem after ldquoThe def-inition also ensures that () implies ()rdquo I haveadded the clause ldquolet (x y) = (c d) in ()rdquo

In secta) the displayed proportion (a b) amp (b c) a c is

no longer numberedb) I have deleted the second ldquothisrdquo in the garbled sen-

tence ldquoDescartes shows this implicitly this in orderto solve ancient problemsrdquo

In secta) at the end of the first paragraph the reference to

ldquop rdquo now has the proper spacing (the TEX codeis now p284 but I had forgotten the backslashbefore)

b) in the second paragraph in the clause that was ldquothepoint now designed by (x y) is the one that wascalled (x minus hya y) beforerdquo I have corrected ldquode-signedrdquo to ldquodesignatedrdquo and made the minus sign aplus sign

In secta) I have enlarged and expanded Figure b) in the fourth paragraph the clause beginning ldquoIf d

is the distance between the new originrdquo continuesnow with ldquoandrdquo (not ldquotordquo)

c) in the fifth paragraph the letter Β at the end is nowthus a Beta rather than B

d) in the sixth paragraph after the beginning ldquoA lengthΗrdquo the parenthesis ldquonot depictedrdquo is added

e) the proportion () formerly had the same internallabel as () which meant that the ensuing proof

of () was wrongly justified by a reference to itselfrather than to ()

In the third paragraph of Chapter ldquodirectionrdquo has thusbeen made singular

Contents

Introduction

The Nine-point Circle

Centers of a triangle The angle in the semicircle The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem Thales and Desargues Locus problems The nine-point conic

Lengths and Areas

Algebra Apollonius

Unity

Bibliography

List of Figures

The nine points A nine-point hyperbola A nine-point ellipse

The arbelos Concurrence of altitudes Bisector of two sides of a triangle Concurrence of medians The angle in the semicircle Right angles and circles The nine-point circle

Elements i The rectangular case of Thalesrsquos Theorem Intermediate form of Thalesrsquos Theorem Converse of Elements i Thalesrsquos Theorem Transitivity Desarguesrsquos Theorem The quadratrix Solution of a five-line locus problem

Hilbertrsquos multiplication Pappusrsquos Theorem Hilbertrsquos trigonometry

Associativity and commutativity Distributivity Circle inscribed in triangle Area of triangle in two ways Change of coordinates Proposition i of Apollonius Proposition i of Apollonius

Symmetries

List of Figures

Introduction

According to Herodotus of Halicarnassus a war was ended bya solar eclipse and Thales of Miletus had predicted the year[ i] The war was between the Lydians and the Medes inAsia Minor the year was the sixth of the war The eclipse isthought to be the one that must have occurred on May ofthe Julian calendar in the year before the Common Era[ p n ]

The birth of Thales is sometimes assigned to the year bce This is done in the ldquoThalesrdquo article on Wikipedia []but it was also done in ancient times (according to the reck-oning of years by Olympiads) The sole reason for this assign-ment seems to be the assumption that Thales must have beenforty when he predicted the eclipse [ p n ]

The nine-point circle was found early in the nineteenth cen-tury of the Common Era The -point conic a generalizationwas found later in the century The existence of the curvesis proved with mathematics that can be traced to Thales andthat is learned today in high school

In the Euclidean plane every triangle determines a fewpoints individually the incenter circumcenter orthocenterand centroid In addition to the vertices themselves the tri-angle also determines various triples of points such as thosein Figure the feet bull of the altitudes the midpoints ofthe sides and the midpoints N of the straight lines runningfrom the orthocenter to the three vertices It turns out that

b

b

b

l

ll

u

u u

Figure The nine points

these nine points lie on a circle called the nine-point circle

or Euler circle of the triangle The discovery of this circleseems to have been published first by Brianchon and Ponceletin ndash [] then by Feuerbach in [ ]

Precise references for the discovery of the nine-point circleare given in Boyerrsquos History of Mathematics [ pp ndash]and Klinersquos Mathematical Thought from Ancient to Modern

Times [ p ] However as with the birth of Thalesprecision is different from accuracy Kline attributes the firstpublication on the nine-point circle to ldquoGergonne and Pon-celetrdquo In consulting his notes Kline may have confused anauthor with the publisher who was himself a mathematicianBoyer mentions that the joint paper of Brianchon and Ponceletwas published in Gergonnersquos Annales The confusion here is areminder that even seemingly authoritative sources may be inerror

The mathematician is supposed to be a skeptic accepting

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

ence to Elements i I have added notice of Proclusrsquosattribution to Thales and a reference to Chapter

In Figure in sect labelled points of intersection areno longer emphasized with dots

In secta) Towards the end of the first paragraph where the

relation defined by () is observed to be transitiveif we allow passage to a third dimension the originalpassage was to a fourth dimension in which a middot d middote middot f = b middot c middot d middot e

b) in the paragraph before Theorem after ldquoThe def-inition also ensures that () implies ()rdquo I haveadded the clause ldquolet (x y) = (c d) in ()rdquo

In secta) the displayed proportion (a b) amp (b c) a c is

no longer numberedb) I have deleted the second ldquothisrdquo in the garbled sen-

tence ldquoDescartes shows this implicitly this in orderto solve ancient problemsrdquo

In secta) at the end of the first paragraph the reference to

ldquop rdquo now has the proper spacing (the TEX codeis now p284 but I had forgotten the backslashbefore)

b) in the second paragraph in the clause that was ldquothepoint now designed by (x y) is the one that wascalled (x minus hya y) beforerdquo I have corrected ldquode-signedrdquo to ldquodesignatedrdquo and made the minus sign aplus sign

In secta) I have enlarged and expanded Figure b) in the fourth paragraph the clause beginning ldquoIf d

is the distance between the new originrdquo continuesnow with ldquoandrdquo (not ldquotordquo)

c) in the fifth paragraph the letter Β at the end is nowthus a Beta rather than B

d) in the sixth paragraph after the beginning ldquoA lengthΗrdquo the parenthesis ldquonot depictedrdquo is added

e) the proportion () formerly had the same internallabel as () which meant that the ensuing proof

of () was wrongly justified by a reference to itselfrather than to ()

In the third paragraph of Chapter ldquodirectionrdquo has thusbeen made singular

Contents

Introduction

The Nine-point Circle

Centers of a triangle The angle in the semicircle The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem Thales and Desargues Locus problems The nine-point conic

Lengths and Areas

Algebra Apollonius

Unity

Bibliography

List of Figures

The nine points A nine-point hyperbola A nine-point ellipse

The arbelos Concurrence of altitudes Bisector of two sides of a triangle Concurrence of medians The angle in the semicircle Right angles and circles The nine-point circle

Elements i The rectangular case of Thalesrsquos Theorem Intermediate form of Thalesrsquos Theorem Converse of Elements i Thalesrsquos Theorem Transitivity Desarguesrsquos Theorem The quadratrix Solution of a five-line locus problem

Hilbertrsquos multiplication Pappusrsquos Theorem Hilbertrsquos trigonometry

Associativity and commutativity Distributivity Circle inscribed in triangle Area of triangle in two ways Change of coordinates Proposition i of Apollonius Proposition i of Apollonius

Symmetries

List of Figures

Introduction

According to Herodotus of Halicarnassus a war was ended bya solar eclipse and Thales of Miletus had predicted the year[ i] The war was between the Lydians and the Medes inAsia Minor the year was the sixth of the war The eclipse isthought to be the one that must have occurred on May ofthe Julian calendar in the year before the Common Era[ p n ]

The birth of Thales is sometimes assigned to the year bce This is done in the ldquoThalesrdquo article on Wikipedia []but it was also done in ancient times (according to the reck-oning of years by Olympiads) The sole reason for this assign-ment seems to be the assumption that Thales must have beenforty when he predicted the eclipse [ p n ]

The nine-point circle was found early in the nineteenth cen-tury of the Common Era The -point conic a generalizationwas found later in the century The existence of the curvesis proved with mathematics that can be traced to Thales andthat is learned today in high school

In the Euclidean plane every triangle determines a fewpoints individually the incenter circumcenter orthocenterand centroid In addition to the vertices themselves the tri-angle also determines various triples of points such as thosein Figure the feet bull of the altitudes the midpoints ofthe sides and the midpoints N of the straight lines runningfrom the orthocenter to the three vertices It turns out that

b

b

b

l

ll

u

u u

Figure The nine points

these nine points lie on a circle called the nine-point circle

or Euler circle of the triangle The discovery of this circleseems to have been published first by Brianchon and Ponceletin ndash [] then by Feuerbach in [ ]

Precise references for the discovery of the nine-point circleare given in Boyerrsquos History of Mathematics [ pp ndash]and Klinersquos Mathematical Thought from Ancient to Modern

Times [ p ] However as with the birth of Thalesprecision is different from accuracy Kline attributes the firstpublication on the nine-point circle to ldquoGergonne and Pon-celetrdquo In consulting his notes Kline may have confused anauthor with the publisher who was himself a mathematicianBoyer mentions that the joint paper of Brianchon and Ponceletwas published in Gergonnersquos Annales The confusion here is areminder that even seemingly authoritative sources may be inerror

The mathematician is supposed to be a skeptic accepting

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

is the distance between the new originrdquo continuesnow with ldquoandrdquo (not ldquotordquo)

c) in the fifth paragraph the letter Β at the end is nowthus a Beta rather than B

d) in the sixth paragraph after the beginning ldquoA lengthΗrdquo the parenthesis ldquonot depictedrdquo is added

e) the proportion () formerly had the same internallabel as () which meant that the ensuing proof

of () was wrongly justified by a reference to itselfrather than to ()

In the third paragraph of Chapter ldquodirectionrdquo has thusbeen made singular

Contents

Introduction

The Nine-point Circle

Centers of a triangle The angle in the semicircle The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem Thales and Desargues Locus problems The nine-point conic

Lengths and Areas

Algebra Apollonius

Unity

Bibliography

List of Figures

The nine points A nine-point hyperbola A nine-point ellipse

The arbelos Concurrence of altitudes Bisector of two sides of a triangle Concurrence of medians The angle in the semicircle Right angles and circles The nine-point circle

Elements i The rectangular case of Thalesrsquos Theorem Intermediate form of Thalesrsquos Theorem Converse of Elements i Thalesrsquos Theorem Transitivity Desarguesrsquos Theorem The quadratrix Solution of a five-line locus problem

Hilbertrsquos multiplication Pappusrsquos Theorem Hilbertrsquos trigonometry

Associativity and commutativity Distributivity Circle inscribed in triangle Area of triangle in two ways Change of coordinates Proposition i of Apollonius Proposition i of Apollonius

Symmetries

List of Figures

Introduction

According to Herodotus of Halicarnassus a war was ended bya solar eclipse and Thales of Miletus had predicted the year[ i] The war was between the Lydians and the Medes inAsia Minor the year was the sixth of the war The eclipse isthought to be the one that must have occurred on May ofthe Julian calendar in the year before the Common Era[ p n ]

The birth of Thales is sometimes assigned to the year bce This is done in the ldquoThalesrdquo article on Wikipedia []but it was also done in ancient times (according to the reck-oning of years by Olympiads) The sole reason for this assign-ment seems to be the assumption that Thales must have beenforty when he predicted the eclipse [ p n ]

The nine-point circle was found early in the nineteenth cen-tury of the Common Era The -point conic a generalizationwas found later in the century The existence of the curvesis proved with mathematics that can be traced to Thales andthat is learned today in high school

In the Euclidean plane every triangle determines a fewpoints individually the incenter circumcenter orthocenterand centroid In addition to the vertices themselves the tri-angle also determines various triples of points such as thosein Figure the feet bull of the altitudes the midpoints ofthe sides and the midpoints N of the straight lines runningfrom the orthocenter to the three vertices It turns out that

b

b

b

l

ll

u

u u

Figure The nine points

these nine points lie on a circle called the nine-point circle

or Euler circle of the triangle The discovery of this circleseems to have been published first by Brianchon and Ponceletin ndash [] then by Feuerbach in [ ]

Precise references for the discovery of the nine-point circleare given in Boyerrsquos History of Mathematics [ pp ndash]and Klinersquos Mathematical Thought from Ancient to Modern

Times [ p ] However as with the birth of Thalesprecision is different from accuracy Kline attributes the firstpublication on the nine-point circle to ldquoGergonne and Pon-celetrdquo In consulting his notes Kline may have confused anauthor with the publisher who was himself a mathematicianBoyer mentions that the joint paper of Brianchon and Ponceletwas published in Gergonnersquos Annales The confusion here is areminder that even seemingly authoritative sources may be inerror

The mathematician is supposed to be a skeptic accepting

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Contents

Introduction

The Nine-point Circle

Centers of a triangle The angle in the semicircle The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem Thales and Desargues Locus problems The nine-point conic

Lengths and Areas

Algebra Apollonius

Unity

Bibliography

List of Figures

The nine points A nine-point hyperbola A nine-point ellipse

The arbelos Concurrence of altitudes Bisector of two sides of a triangle Concurrence of medians The angle in the semicircle Right angles and circles The nine-point circle

Elements i The rectangular case of Thalesrsquos Theorem Intermediate form of Thalesrsquos Theorem Converse of Elements i Thalesrsquos Theorem Transitivity Desarguesrsquos Theorem The quadratrix Solution of a five-line locus problem

Hilbertrsquos multiplication Pappusrsquos Theorem Hilbertrsquos trigonometry

Associativity and commutativity Distributivity Circle inscribed in triangle Area of triangle in two ways Change of coordinates Proposition i of Apollonius Proposition i of Apollonius

Symmetries

List of Figures

Introduction

According to Herodotus of Halicarnassus a war was ended bya solar eclipse and Thales of Miletus had predicted the year[ i] The war was between the Lydians and the Medes inAsia Minor the year was the sixth of the war The eclipse isthought to be the one that must have occurred on May ofthe Julian calendar in the year before the Common Era[ p n ]

The birth of Thales is sometimes assigned to the year bce This is done in the ldquoThalesrdquo article on Wikipedia []but it was also done in ancient times (according to the reck-oning of years by Olympiads) The sole reason for this assign-ment seems to be the assumption that Thales must have beenforty when he predicted the eclipse [ p n ]

The nine-point circle was found early in the nineteenth cen-tury of the Common Era The -point conic a generalizationwas found later in the century The existence of the curvesis proved with mathematics that can be traced to Thales andthat is learned today in high school

In the Euclidean plane every triangle determines a fewpoints individually the incenter circumcenter orthocenterand centroid In addition to the vertices themselves the tri-angle also determines various triples of points such as thosein Figure the feet bull of the altitudes the midpoints ofthe sides and the midpoints N of the straight lines runningfrom the orthocenter to the three vertices It turns out that

b

b

b

l

ll

u

u u

Figure The nine points

these nine points lie on a circle called the nine-point circle

or Euler circle of the triangle The discovery of this circleseems to have been published first by Brianchon and Ponceletin ndash [] then by Feuerbach in [ ]

Precise references for the discovery of the nine-point circleare given in Boyerrsquos History of Mathematics [ pp ndash]and Klinersquos Mathematical Thought from Ancient to Modern

Times [ p ] However as with the birth of Thalesprecision is different from accuracy Kline attributes the firstpublication on the nine-point circle to ldquoGergonne and Pon-celetrdquo In consulting his notes Kline may have confused anauthor with the publisher who was himself a mathematicianBoyer mentions that the joint paper of Brianchon and Ponceletwas published in Gergonnersquos Annales The confusion here is areminder that even seemingly authoritative sources may be inerror

The mathematician is supposed to be a skeptic accepting

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

List of Figures

The nine points A nine-point hyperbola A nine-point ellipse

The arbelos Concurrence of altitudes Bisector of two sides of a triangle Concurrence of medians The angle in the semicircle Right angles and circles The nine-point circle

Elements i The rectangular case of Thalesrsquos Theorem Intermediate form of Thalesrsquos Theorem Converse of Elements i Thalesrsquos Theorem Transitivity Desarguesrsquos Theorem The quadratrix Solution of a five-line locus problem

Hilbertrsquos multiplication Pappusrsquos Theorem Hilbertrsquos trigonometry

Associativity and commutativity Distributivity Circle inscribed in triangle Area of triangle in two ways Change of coordinates Proposition i of Apollonius Proposition i of Apollonius

Symmetries

List of Figures

Introduction

According to Herodotus of Halicarnassus a war was ended bya solar eclipse and Thales of Miletus had predicted the year[ i] The war was between the Lydians and the Medes inAsia Minor the year was the sixth of the war The eclipse isthought to be the one that must have occurred on May ofthe Julian calendar in the year before the Common Era[ p n ]

The birth of Thales is sometimes assigned to the year bce This is done in the ldquoThalesrdquo article on Wikipedia []but it was also done in ancient times (according to the reck-oning of years by Olympiads) The sole reason for this assign-ment seems to be the assumption that Thales must have beenforty when he predicted the eclipse [ p n ]

The nine-point circle was found early in the nineteenth cen-tury of the Common Era The -point conic a generalizationwas found later in the century The existence of the curvesis proved with mathematics that can be traced to Thales andthat is learned today in high school

In the Euclidean plane every triangle determines a fewpoints individually the incenter circumcenter orthocenterand centroid In addition to the vertices themselves the tri-angle also determines various triples of points such as thosein Figure the feet bull of the altitudes the midpoints ofthe sides and the midpoints N of the straight lines runningfrom the orthocenter to the three vertices It turns out that

b

b

b

l

ll

u

u u

Figure The nine points

these nine points lie on a circle called the nine-point circle

or Euler circle of the triangle The discovery of this circleseems to have been published first by Brianchon and Ponceletin ndash [] then by Feuerbach in [ ]

Precise references for the discovery of the nine-point circleare given in Boyerrsquos History of Mathematics [ pp ndash]and Klinersquos Mathematical Thought from Ancient to Modern

Times [ p ] However as with the birth of Thalesprecision is different from accuracy Kline attributes the firstpublication on the nine-point circle to ldquoGergonne and Pon-celetrdquo In consulting his notes Kline may have confused anauthor with the publisher who was himself a mathematicianBoyer mentions that the joint paper of Brianchon and Ponceletwas published in Gergonnersquos Annales The confusion here is areminder that even seemingly authoritative sources may be inerror

The mathematician is supposed to be a skeptic accepting

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Associativity and commutativity Distributivity Circle inscribed in triangle Area of triangle in two ways Change of coordinates Proposition i of Apollonius Proposition i of Apollonius

Symmetries

List of Figures

Introduction

According to Herodotus of Halicarnassus a war was ended bya solar eclipse and Thales of Miletus had predicted the year[ i] The war was between the Lydians and the Medes inAsia Minor the year was the sixth of the war The eclipse isthought to be the one that must have occurred on May ofthe Julian calendar in the year before the Common Era[ p n ]

The birth of Thales is sometimes assigned to the year bce This is done in the ldquoThalesrdquo article on Wikipedia []but it was also done in ancient times (according to the reck-oning of years by Olympiads) The sole reason for this assign-ment seems to be the assumption that Thales must have beenforty when he predicted the eclipse [ p n ]

The nine-point circle was found early in the nineteenth cen-tury of the Common Era The -point conic a generalizationwas found later in the century The existence of the curvesis proved with mathematics that can be traced to Thales andthat is learned today in high school

In the Euclidean plane every triangle determines a fewpoints individually the incenter circumcenter orthocenterand centroid In addition to the vertices themselves the tri-angle also determines various triples of points such as thosein Figure the feet bull of the altitudes the midpoints ofthe sides and the midpoints N of the straight lines runningfrom the orthocenter to the three vertices It turns out that

b

b

b

l

ll

u

u u

Figure The nine points

these nine points lie on a circle called the nine-point circle

or Euler circle of the triangle The discovery of this circleseems to have been published first by Brianchon and Ponceletin ndash [] then by Feuerbach in [ ]

Precise references for the discovery of the nine-point circleare given in Boyerrsquos History of Mathematics [ pp ndash]and Klinersquos Mathematical Thought from Ancient to Modern

Times [ p ] However as with the birth of Thalesprecision is different from accuracy Kline attributes the firstpublication on the nine-point circle to ldquoGergonne and Pon-celetrdquo In consulting his notes Kline may have confused anauthor with the publisher who was himself a mathematicianBoyer mentions that the joint paper of Brianchon and Ponceletwas published in Gergonnersquos Annales The confusion here is areminder that even seemingly authoritative sources may be inerror

The mathematician is supposed to be a skeptic accepting

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Introduction

According to Herodotus of Halicarnassus a war was ended bya solar eclipse and Thales of Miletus had predicted the year[ i] The war was between the Lydians and the Medes inAsia Minor the year was the sixth of the war The eclipse isthought to be the one that must have occurred on May ofthe Julian calendar in the year before the Common Era[ p n ]

The birth of Thales is sometimes assigned to the year bce This is done in the ldquoThalesrdquo article on Wikipedia []but it was also done in ancient times (according to the reck-oning of years by Olympiads) The sole reason for this assign-ment seems to be the assumption that Thales must have beenforty when he predicted the eclipse [ p n ]

The nine-point circle was found early in the nineteenth cen-tury of the Common Era The -point conic a generalizationwas found later in the century The existence of the curvesis proved with mathematics that can be traced to Thales andthat is learned today in high school

In the Euclidean plane every triangle determines a fewpoints individually the incenter circumcenter orthocenterand centroid In addition to the vertices themselves the tri-angle also determines various triples of points such as thosein Figure the feet bull of the altitudes the midpoints ofthe sides and the midpoints N of the straight lines runningfrom the orthocenter to the three vertices It turns out that

b

b

b

l

ll

u

u u

Figure The nine points

these nine points lie on a circle called the nine-point circle

or Euler circle of the triangle The discovery of this circleseems to have been published first by Brianchon and Ponceletin ndash [] then by Feuerbach in [ ]

Precise references for the discovery of the nine-point circleare given in Boyerrsquos History of Mathematics [ pp ndash]and Klinersquos Mathematical Thought from Ancient to Modern

Times [ p ] However as with the birth of Thalesprecision is different from accuracy Kline attributes the firstpublication on the nine-point circle to ldquoGergonne and Pon-celetrdquo In consulting his notes Kline may have confused anauthor with the publisher who was himself a mathematicianBoyer mentions that the joint paper of Brianchon and Ponceletwas published in Gergonnersquos Annales The confusion here is areminder that even seemingly authoritative sources may be inerror

The mathematician is supposed to be a skeptic accepting

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

b

b

b

l

ll

u

u u

Figure The nine points

these nine points lie on a circle called the nine-point circle

or Euler circle of the triangle The discovery of this circleseems to have been published first by Brianchon and Ponceletin ndash [] then by Feuerbach in [ ]

Precise references for the discovery of the nine-point circleare given in Boyerrsquos History of Mathematics [ pp ndash]and Klinersquos Mathematical Thought from Ancient to Modern

Times [ p ] However as with the birth of Thalesprecision is different from accuracy Kline attributes the firstpublication on the nine-point circle to ldquoGergonne and Pon-celetrdquo In consulting his notes Kline may have confused anauthor with the publisher who was himself a mathematicianBoyer mentions that the joint paper of Brianchon and Ponceletwas published in Gergonnersquos Annales The confusion here is areminder that even seemingly authoritative sources may be inerror

The mathematician is supposed to be a skeptic accepting

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

nothing before knowing its proof In practice this does notalways happen even in mathematics But the ideal shouldbe maintained even in subjects other than mathematics likehistory

There is however another side to this ideal We shall referoften in this essay to Euclidrsquos Elements [ ] andespecially to the first of its thirteen books Euclid is accused ofaccepting things without proof Two sections of Klinersquos historyare called ldquoThe Merits and Defects of the Elementsrdquo [ ch sect p ] and ldquoThe Defects in Euclidrdquo (ch sect p )One of the supposed ldquodefectsrdquo is

he uses dozens of assumptions that he never states and un-doubtedly did not recognize

We all do this all the time and it is not a defect We cannotstate everything that we assume even the possibility of statingthings is based on assumptions about language itself We tryto state some of our assumptions in order to question themas when we encounter a problem with the ordinary course oflife By the account of R G Collingwood [] the attempt towork out our fundamental assumptions is metaphysics

Herodotus says the Greeks learned geometry from the Egyp-tians who needed it in order to measure how much landthey lost to the flooding of the Nile Herodotusrsquos word γε-

ωμετρίη means also surveying or ldquothe art of measuring landrdquo[ ii] According to David Fowler in The Mathematics

of Platorsquos Academy [ sect(d) pp ndash sect pp ndash]the Egyptians defined the area of a quadrilateral field as theproduct of the averages of the pairs of opposite sides

The Egyptian rule is not strictly accurate Book i of theElements corrects the error The climax of the book is thedemonstration in Proposition that every straight-sidedfield is exactly equal to a certain parallelogram with a given

Introduction

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

side

Euclidrsquos demonstrations take place in a world where asArchimedes postulates [ p ]

among unequal [magnitudes] the greater exceeds the smallerby such [a difference] that is capable added itself to itselfof exceeding everything set forth

This is the world in which the theory of proportion in Bookv of the Elements is valid A theory of proportion is neededfor the Cartesian geometry whereby the nine-point conic isestablished

One can develop a theory of proportion that does not requirethe Archimedean assumption Is it a defect that Euclid doesnot do this We shall do it trying to put into the theory onlyenough to make Thalesrsquos Theorem true This is the theoremthat if a straight line cuts two sides of a triangle it cuts themproportionally if and only if it is parallel to the third side Ishall call this Thalesrsquos Theorem for convenience and becauseit is so called in some countries today [] There is also somehistorical basis for the name we shall investigate how much

The present essay is based in part on notes prepared origi-nally for one of several twenty-minute talks at the Thales Bu-

luşması (Thales Meeting) held in the Roman theater in theruins of Thalesrsquos home town September The eventwas arranged by the Tourism Research Society (Turizm Araştır-

maları Derneği TURAD) and the office of the mayor of DidimPart of the Aydın province of Turkey the district of Didimencompasses the ancient Ionian cities of Priene and Miletusalong with the temple of Didyma This temple was linked toMiletus and Herodotus refers to the temple under the nameof the family of priests the Branchidae

My essay is based also on notes from a course on Pappusrsquos

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Theorem and projective geometry given at my home univer-sity Mimar Sinan in Istanbul and at the Nesin MathematicsVillage near the Ionian city of Ephesus

To seal the Peace of the Eclipse the Lydian King Alyat-tes gave his daughter Aryenis to Astyages son of the MedianKing Cyaxares [ i] It is not clear whether Aryenis wasthe mother of Astyagesrsquos daughter Mandane whom Astyagesmarried to the Persian Cambyses and whose son Astyagestried to murder because of the Magirsquos unfavorable interpre-tation of certain dreams [ indash] That son was Cyruswho survived and grew up to conquer his grandfather AgainHerodotus is not clear that this was the reason for the quarrelwith Cyrus by Croesus [ i] who was son and successor ofAlyattes and thus brother of Astyagesrsquos consort Aryenis ButCroesus was advised by the oracles at Delphi and Amphiarausthat if he attacked Persia a great empire would be destroyedand that he should make friends with the mightiest of theGreeks [ indash] Perhaps it was in obedience to this ora-cle that Croesus sought the alliance with Miletus mentionedby Diogenes Laertius who reports that Thales frustrated theplan and ldquothis proved the salvation of the city when Cyrusobtained the victoryrdquo [ i] Nonetheless Herodotus re-ports a general Greek beliefmdashwhich he does not acceptmdashthatThales helped Croesusrsquos army march to Persia by divertingthe River Halys (todayrsquos Kızılırmak) around them [ i]But Croesus was defeated and thus his own great empire wasdestroyed

When the victorious Cyrus returned east from the Lydiancapital of Sardis he left behind a Persian called Tabulus torule but a Lydian called Pactyes to be treasurer [ i]Pactyes mounted a rebellion but it failed and he sought asy-lum in the Aeolian city of Cyme The Cymaeans were told by

Introduction

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

the oracle at Didyma to give him up [ indash] In disbeliefa Cymaean called Aristodicus began driving away the birdsthat nested around the temple

But while he so did a voice (they say) came out of the innershrine calling to Aristodicus and saying ldquoThou wickedest ofmen wherefore darest thou do this wilt thou rob my templeof those that take refuge with merdquo Then Aristodicus had hisanswer ready ldquoO Kingrdquo said he ldquowilt thou thus save thineown suppliants yet bid the men of Cyme deliver up theirsrdquoBut the god made answer ldquoYea I do bid them that ye maythe sooner perish for your impiety and never again come toinquire of my oracle concerning the giving up of them thatseek refuge with yourdquo

As the temple survives today so does the sense of the injunc-tion of the oracle in a Turkish saying [ p ]

İsteyenin bir yuumlzuuml kara vermeyenin iki yuumlzuumlWho asks has a black face but who does not give has two

From his studies of art history and philosophy Colling-wood concluded that ldquoall history is the history of thoughtrdquo [p ] As a form of thought mathematics has a history Un-fortunately this is forgotten in some Wikipedia articles wheredefinitions and results may be laid out as if they have been un-derstood since the beginning of time We can all rectify thissituation if we will by contributing to the encyclopedia OnMay to the article ldquoPappusrsquos Hexagon Theoremrdquo []I added a section called ldquoOriginsrdquo giving Pappusrsquos own proofThe theorem can be seen as lying behind Cartesian geometry

In his Geometry of Descartes takes inspiration fromPappus whom he quotes in Latin presumably from Com-mandinusrsquos edition of [ p n ] the Frenchedition of the Geometry has a footnote [ p ] seemingly inDescartesrsquos voice although other footnotes are obviously from

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

an editor ldquoI cite rather the Latin version than the Greek textso that everybody will understand me more easilyrdquo

The admirable Princeton Companion to Mathematics [pp ndash] says a lot about where mathematics is now in itshistory In one chapter editor Timothy Gowers discusses ldquoTheGeneral Goals of Mathematical Researchrdquo He divides thesegoals among nine sections () Solving Equations () Clas-sifying () Generalizing () Discovering Patterns () Ex-plaining Apparent Coincidences () Counting and Measuring() Determining Whether Different Mathematical PropertiesAre Compatible () Working with Arguments That Are NotFully Rigorous () Finding Explicit Proofs and AlgorithmsThese are some goals of research today There is a tenth sec-tion of the chapter but its title is general ldquoWhat Do You Findin a Mathematical Paperrdquo As Gowers says the kind of paperthat he means is one written on a pattern established in thetwentieth century

The Princeton Companion is expressly not an encyclopediaOne must not expect every species of mathematics to meet oneor more of Gowersrsquos enumerated goals Geometrical theoremslike that of the nine-point circle do not really seem to meetthe goals They are old-fashioned The nine-point circle itselfis not the explanation of a coincidence it is the coincidencethat a certain set of nine points all happen to lie at the samedistance from a tenth point A proof of this coincidence maybe all the explanation there is The proof might be describedas explicit in the sense of showing how that tenth point canbe found but in this case there can be no other kind of proofFrom any three points of a circle the center can be found

One can generalize the nine-point circle obtaining the nine-

point conic which is determined by any four points in theEuclidean plane provided no three are collinear As in Fig-

Introduction

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

A B

C

D

b

b

b

uu

u

u

uu

Figure A nine-point hyperbola

ures and where the four points are A B C and Dthe conic passes through the midpoints N of the straight linesbounded by the six pairs formed out of the four points andit also passes through the intersection points bull of the pairs ofstraight lines that together pass through all four points

The discovery of the nine-point circle itself would seemnot to be the accomplishment of any particular goal beyondsimple enjoyment Indeed one might make an alternativelist of goals of mathematical research () personal satisfac-tion () satisfying collaboration with friends and colleagues() impressing those friends and colleagues () serving scienceand industry () winning a grant () earning a promotion() finding a job in the first place These may be goals in anyacademic pursuit But none of them can come to be recog-nized as goals unless the first one or two have actually been

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

A

B

C

D

bb

bu

u

u

u

u

u

Figure A nine-point ellipse

achieved First you have to find out by chance that somethingis worth doing for its own sake before you can put it to someother use

The present essay is an illustration of the general point Afellow alumnus of St Johnrsquos College [] expressed to me alongwith other alumni and alumnae the pleasure of learning thenine-point circle Not having done so before I learned it tooand also the nine-point conic I wrote out proofs for my ownsatisfaction The proof of the circle uses the theorem that astraight line bisecting two sides of a triangle is parallel to thethird This is a special case of Thalesrsquos Theorem The attribu-tion to Thales actually obscures some interesting mathematicsso I started writing about this using the notes I mentionedThalesrsquos Theorem allows a theoretical justification of the mul-tiplication that Descartes defines in order to introduce algebra

Introduction

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

to geometry The nine-point conic is an excellent illustrationof the power of Descartesrsquos geometry Thalesrsquos Theorem canmake the geometry rigorous However David Hilbert takesanother approach

The first-year students in my department in Istanbul readEuclid for themselves in their first semester They learn im-plicitly about Descartes in their second semester in lectureson analytic geometry I have read Pappus with older studentstoo as I mentioned All of the courses have been an oppor-tunity and an impetus to clarify the transition from Euclid toDescartes The nine-point circle and conic provide an occasionto bring the ideas together but doing this does not representthe accomplishment of a preconceived goal

The high-school geometry course that I took in ndash inWashington could have included the nine-point circle Thecourse was based on the text by Weeks and Adkins who taughtproofs in the two-column ldquostatementndashreasonrdquo format [ ppndash] A edition of the text is apparently still availablealbeit from a small publisher The persistence of the text issatisfying though I was not satisfied by the book as a studentThe tedious style had me wondering why we did not just readEuclidrsquos Elements I did this on my own and I did it threeyears later as a student at St Johnrsquos College

After experiencing Euclid both as student and teacherI have gone back to detect foundational weaknesses in theWeeksndashAdkins text One of them is the confusion of equal-ity with sameness I discuss this in detail elsewhere [] Thedistinction between equality and sameness is important in ge-ometry if not in algebra In geometry equal line segments havethe same length but the line segments are still not in any sensethe same segment An isosceles triangle has two equal sidesBut in Euclid two ratios are never equal although they may

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

be the same This helps clarify what can be meant by a ratioin the first place In modern terms a ratio is an equivalenceclass so any definition of ratio must respect this

Another weakness of Weeks and Adkins has been shared bymost modern books as far as I know since Descartesrsquos Geom-etry The weakness lies in the treatment of Thalesrsquos TheoremThe meaning of the theorem the truth of the theorem andthe use of the theorem to justify algebramdashnone of these areobvious In The Foundations of Geometry Hilbert recognizesthe need for work here and he does the work [ pp ndash]Weeks and Adkins recognize the need too but only to the ex-tent of proving Thalesrsquos Theorem for commensurable divisionsthen mentioning that there is an incommensurable case Theydo this in a section labelled [b] for difficulty and omissibility[ pp v ndash] They say

Ideas involved in proofs of theorems for incommensurablesegments are too difficult for this stage of our mathematics

Such condescension is annoying but in any case we shall es-tablish Thalesrsquos Theorem as Hilbert does in the sense of notusing the Archimedean assumption that underlies Euclidrsquos no-tion of commensurability

First we shall use the theory of areas as developed in Book i

of the Elements This relies on Common Notion the wholeis greater than the part not only when the whole is a boundedstraight line but also when it is a bounded region of the planeWhen point B lies between A and C on a straight line thenAC is greater than AB and when two rectangles share a basethe rectangle with the greater height is the greater Hilbertshows how to prove the latter assertion from the former Hedoes this by developing an ldquoalgebra of segmentsrdquo

We shall review this ldquoalgebra of segmentsrdquo but first we shallfocus on the ldquoalgebra of areasrdquo It is not really algebra in the

Introduction

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

sense of relying on strings of juxtaposed symbols it relies onan understanding of pictures In The Shaping of Deduction

in Greek Mathematics [ p ] Reviel Netz examines howthe diagram of an ancient Greek mathematical proposition isnot always recoverable from the text alone The diagram isan integral part of the proposition even its ldquometonymrdquo itstands for the entire proposition the way the enunciation ofthe proposition stands today [ p ] In the summary ofEuclid called the Bones [] both the enunciations and thediagrams of the propositions of the Elements are given Unlikesay Homerrsquos Iliad Euclidrsquos Elements is not a work that oneunderstands through hearing it recited by a blind poet

The same will be true for the present essay if only becauseI have not wanted to take the trouble to write out everythingin words The needs of blind readers should be respected butthis might be done best with tactile diagrams which couldbenefit sighted readers as well What Collingwood writes inThe Principles of Art [ pp ndash] about painting appliesalso to mathematics

The forgotten truth about painting which was rediscoveredby what may be called the CeacutezannendashBerenson approach to itwas that the spectatorrsquos experience on looking at a pictureis not a specifically visual experience at all It does notbelong to sight alone it belongs also (and on some occasionseven more essentially) to touch [Berenson] is thinkingor thinking in the main of distance and space and massnot of touch sensations but of motor sensations such as weexperience by using our muscles and moving our limbs Butthese are not actual motor sensations they are imaginarymotor sensations In order to enjoy them when looking ata Masaccio we need not walk straight through the pictureor even stride about the gallery what we are doing is toimagine ourselves as moving in these ways

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

To imagine these movements I would add one needs someexperience of making them Doing mathematics requires somekind of imaginative understanding of what the mathematics isabout This understanding may be engendered by drawingsof triangles and circles but then it might just as well if notbetter be engendered by triangles and circles that can be heldand manipulated

Descartes develops a kind of mathematics that might seemto require a minimum of imagination If you have no idea ofthe points that you are looking for you can just call them(x y) and proceed Pappus describes the general method [ p ]

Now analysis is the path from what one is seeking as if itwere established by way of its consequences to somethingthat is established by synthesis That is to say in analysis weassume what is sought as if it has been achieved and look forthe thing from which it follows and again what comes beforethat until by regressing in this way we come upon some oneof the things that are already known or that occupy the rankof a first principle We call this kind of method lsquoanalysisrsquo asif to say anapalin lysis (reduction backward)

The derivation of the nine-point conic will be by Cartesiananalysis

In Rule Four of the posthumously published Rules for theDirection of the Mind [ p ] Descartes writes of amethod that is

so useful that without it the pursuit of learning would Ithink be more harmful than profitable Hence I can readilybelieve that the great minds of the past were to some ex-tent aware of it guided to it even by nature alone Thisis our experience in the simplest of sciences arithmetic andgeometry we are well aware that the geometers of antiquity

Introduction

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

employed a sort of analysis which they went on to apply tothe solution of every problem though they begrudged reveal-ing it to posterity At the present time a sort of arithmeticcalled ldquoalgebrardquo is flourishing and this is achieving for num-bers what the ancients did for figures But if one attendsclosely to my meaning one will readily see that ordinarymathematics is far from my mind here that it is quite an-other discipline I am expounding and that these illustrationsare more its outer garments than its inner parts Indeedone can even see some traces of this true mathematics Ithink in Pappus and Diophantus who though not of thatearliest antiquity lived many centuries before our time ButI have come to think that these writers themselves with akind of pernicious cunning later suppressed this mathemat-ics as notoriously many inventors are known to have donewhere their own discoveries are concerned In the presentage some very gifted men have tried to revive this methodfor the method seems to me to be none other than the artwhich goes by the outlandish name of ldquoalgebrardquomdashor at leastit would be if algebra were divested of the multiplicity ofnumbers and imprehensible figures which overwhelm it andinstead possessed that abundance of clarity and simplicitywhich I believe true mathematics ought to have

Possibly Apollonius is for Descartes of ldquoearliest antiquityrdquobut in any case he precedes Pappus and Diophantus by cen-turies He may have a secret weapon in coming up with hispropositions about conic sections but pace Descartes I do notthink it is Cartesian analysis One cannot have a method forfinding things unless one already hasmdashor somebody hasmdashagood idea of what one wants to find in the first place As ifopening boxes to see what is inside Apollonius slices conesThis is why we can now write down equations and call themconic sections

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Today we think of conic sections as having axes one forthe parabola and two each for the ellipse and hyperbola Thenotion comes from Apollonius but for him an axis is justa special case of a diameter A diameter of a conic sectionbisects certain chords of the section that are all parallel to oneanother In Book i of the Conics [ ] Apollonius shows thatevery straight line through the center of an ellipse or hyperbolais a diameter in this sense and every straight line parallel tothe axis is a diameter of a parabola One can give a proofby formal change of coordinates but the proof of Apolloniusinvolves areas and it does not seem likely that this is histranslation of the former proof In any case for comparisonwe shall set down both proofs for the parabola at least

In introducing the nine-point circle near the beginning of hisIntroduction to Geometry [ p ] Coxeter quotes Pedoe onthe same subject from Circles [ p ]

This [nine-point] circle is the first really exciting one to ap-pear in any course on elementary geometry

I am not sure whether to read this as encouragement to learnthe nine-point circle or as disparagement of the education thatthe student might have had to endure in order to be able tolearn the circle In any case all Euclidean circles are the samein isolation In Book iii of the Elements are the theorems thatevery angle in a semicircle is right (iii) and that the parts ofintersecting chords of a circle bound equal rectangles (iii)The former theorem is elsewhere attributed to Thales Donot both theorems count as exciting The nine-point circle isexciting for combining the triangles of Book i with the circlesof Book iii

Introduction

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

The Nine-point Circle

Centers of a triangle

The angle bisectors of a triangle and the perpendicular bi-sectors of the sides of the triangle meet respectively at singlepoints called today the incenter and circumcenter of thetriangle [ pp ndash] This is an implicit consequence ofElements iv and where circles are respectively inscribedin and circumscribed about a triangle the centers of thesecircles are the points just mentioned

The concurrence of the altitudes of a triangle is used inthe Book of Lemmas The book is attributed ultimately toArchimedes and Heiberg includes a Latin rendition in his ownedition of Archimedes [ p ] However the book comesdown to us originally in Arabic Its Proposition [ p ndash] concerns a semicircle with two semicircles removed as inFigure the text quotes Archimedes as having called theshape an arbelos or shoemakerrsquos knife In the only suchinstance that I know of the big LiddellndashScott lexicon [ p] illustrates the ἄρβηλος entry with a picture of the shapeThe term and the shape came to my attention before the high-school course that I mentioned in a ldquoMathematical Gamesrdquocolumn of Martin Gardner [ ch p ] The secondtheorem that Gardner mentions is Proposition of the Book

of Lemmas the arbelos ABCD is equal to the circle whose

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

A B C

D

Figure The arbelos

diameter is BD Proposition is that the circles inscribed inthe two parts into which the arbelos is cut by BD are equalthe proof appeals to the theorem that the altitudes of a triangleconcur at a point

Today that point is the orthocenter of the triangle and itsexistence follows from that of the circumcenter In Figure where the sides of triangle GHK are parallel to the respectivesides of ABC the altitudes AD BE and CF of ABC are

the perpendicular bisectors of the sides of GHK Since theseperpendicular bisectors concur at L so do the altitudes ofABC

In Propositions and of On the Equilibrium of Planes

[ p ndash] Archimedes shows that the center of gravityof a triangle must lie on a median and therefore must lie atthe intersection of two medians Implicitly then the threemedians must concur at a point which we call the centroid

though Archimedesrsquos language suggests that this is known in-dependently The existence of the centroid follows from thespecial case of Thalesrsquos Theorem that we shall want anyway

The Nine-point Circle

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

H

K

G

A B

C

F

DE

L

Figure Concurrence of altitudes

to prove the nine-point circle

Theorem A straight line bisecting a side of a triangle bi-

sects a second side if and only if the cutting line is parallel to

the third side

Proof In triangle ABC in Figure let D be the midpoint ofside AB (Elements i) and let DE and DF be drawn paral-lel to the other sides (Elements i) Then triangles ADE andDBF have equal sides between equal angles (Elements i)so the triangles are congruent (Elements i) In particularAE = DF But in the parallelogram CEDF DF = EC (El-

ements i) Thus AE = EC Therefore DE is the bisectorof two sides of ABC Conversely since there is only one suchbisector it must be the parallel to the third side

For completeness we establish the centroid In Figure

Centers of a triangle

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

A

B C

D E

F

Figure Bisector of two sides of a triangle

if AD and CF are medians of triangle ABC and BH is drawnparallel to AD then by passing though G BE bisects CH bythe theorem just proved and the angles FBH and FAG areequal by Elements i Since the vertical angles BFH andAFG are equal (by Elements i which Proclus attributes toThales as we shall discuss in Chapter ) and BF = AF thetriangles BFH and AFG must be congruent (Elements i)and in particular BH = AG Since these straight lines arealso parallel so are AH and BE (Elements i) Again byTheorem BE bisects AC

The angle in the semicircle

We shall want to know that the angle in a semicircle is rightThis is Proposition iii of Euclid but an attribution toThales is passed along by Diogenes Laertius the biographerof philosophers [ Indash]

Pamphila says that having learnt geometry from the Egyp-tians he [Thales] was the first to inscribe in a circle a right-angled triangle whereupon he sacrificed an ox Others say

The Nine-point Circle

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

bA

b

B

bC

bE

b D

b

F

bG

b

H

≫

≫

Figure Concurrence of medians

it was Pythagoras among them being Apollodorus the cal-culator

The theorem is easily proved by means of Elements i theangles of a triangle are together equal to two right angles Letthe side BC of triangle ABC in Figure be bisected at Dand let AD be drawn If A lies on the circle with diameterBC then the triangles ABD and ACD are isosceles so theirbase angles are equal by Elements i this is also attributedto Thales as we shall discuss later Meanwhile the four baseangles being together equal to two right angles the two of themthat make up angle BAC must together be right The conversefollows from Elements i which has no other obvious usean angle like BEC inscribed in BAC is greater than BACcircumscribed like BFC less

Thales may have established Elements iii but it is hardto attribute to him the proof based on i when Proclus in

The angle in the semicircle

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

B

A

CD

E

F

Figure The angle in the semicircle

his Commentary on the First Book of Euclidrsquos Elements at-tributes this result to the Pythagoreans [ ] who cameafter Thales Proclus cites the now-lost history of geometryby Eudemus who was apparently a student of Aristotle

In his History of Greek Mathematics Heath [ pp ndash]proposes an elaborate argument for iii not using the gen-eral theorem about the sum of the angles of a triangle If arectangle exists one can prove that the diagonals intersect ata point equidistant from the four vertices so that they lie on acircle whose center is that intersection point as in Figure aIn particular then a right angle is inscribed in a semicircle

It seems to me one might just as well draw two diametersof a circle and observe that their endpoints by symmetry arethe vertices of an equiangular quadrilateral This quadrilateralmust then be a rectangle that is the four equal angles ofthe quadrilateral must together make a circle This can beinferred from the observation that equiangular quadrilaterals

The Nine-point Circle

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

b

b

bb

b

(a) A rectangle in a circle (b) A locus of right angles

Figure Right angles and circles

can be used to tile floors

Should the existence of rectangles be counted thus not as atheorem but as an observation if not a postulate In A Short

History of Greek Mathematics which is earlier than Heathrsquoshistory Gow [ p ] passes along a couple of ideas of oneDr Allman about inductive reasoning From floor tiles onemay induce as above that the angle in a semicircle is rightBy observation one may find that the locus of apices of righttriangles whose bases (the hypotenuses) are all the same givensegment is a semicircle as in Figure b

The nine-point circle

Theorem (Nine-point Circle) In any triangle the mid-

points of the sides the feet of the altitudes and the midpoints

of the straight lines drawn from the orthocenter to the vertices

The nine-point circle

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

A

B CD

E

F

P

b

G

b

H

b

K

bL

bM bN

Figure The nine-point circle

lie on a single circle

Proof Suppose the triangle is ABC in Figure Let thealtitudes BE and CF be dropped (Elements i) their inter-section point is P Let AP be drawn and extended as neededso as to meet BC at D We already know that P is the ortho-center of ABC so AD must be at right angles to BC in factwe shall prove this independently

Bisect BC CA and AB at G H and K respectively andbisect PA PB and PC at L M and N respectively By

The Nine-point Circle

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Theorem GK and LN are parallel to AC so they are par-allel to one another by Elements i Similarly KL and NGare parallel to one another being parallel to BE Then thequadrilateral GKLN is a parallelogram it is a rectangle byElements i since AC and BE are at right angles to oneanother Likewise GHLM is a rectangle The two rectangleshave common diagonal GL and so the circle with diameter GLalso passes through the remaining vertices of the rectanglesby the converse of Elements iii discussed above Similarlythe respective diagonals KN and HM of the two rectanglesmust be diameters of the circle and so KHNM is a rectan-gle this yields that AD is at right angles to BC The circlemust pass through E since angle MEH is right and MH isa diameter likewise the circle passes through F and G

The Nine-point Circle Theorem is symmetric in the ver-tices and orthocenter of the triangle These four points havethe property that the straight line through any two of thempasses through neither of the other two moreorever the lineis at right angles to the straight line through the remainingtwo vertices In other words the points are the vertices of acomplete quadrangle and each of its three pairs of oppositesides are at right angles to one another The intersection of apair of opposite sides being called a diagonal point a singlecircle passes through the three of these and the midpoints ofthe six sides

We proceed to the case of a complete quadrangle whoseopposite sides need not be at right angles There will be asingle conic section passing through the three diagonal pointsand the midpoints of the six sides The proof will use Cartesiangeometry as founded on Thalesrsquos Theorem

The nine-point circle

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

The Nine-point Conic

Thalesrsquos Theorem

A rudimentary form of Thalesrsquos Theorem is mentioned in thefanciful dialogue by Plutarch called Dinner of the Seven WiseMen [ sect pp ndash] Here the character of Neiloxenus saysof and to Thales

he does not try to avoid as the rest of you do being a friendof kings and being called such In your case for instancethe king [of Egypt] finds much to admire in you and inparticular he was immensely pleased with your method ofmeasuring the pyramid because without making any adoor asking for any instrument you simply set your walking-stick upright at the edge of the shadow which the pyramidcast and two triangles being formed by the intercepting ofthe sunrsquos rays you demonstrated that the height of the

pyramid bore the same relation to the length of the

stick as the one shadow to the other

The word translated as ldquorelationrdquo here in the sentence that Ihave emboldened is λόγος This is usually translated as ldquoratiordquoin mathematics The whole sentence is

ἔδειξας

ὃν ἡ σκιὰ πρὸς τὴν σκιὰν λόγον εἶχε

τὴν πυραμίδα πρὸς τὴν βακτηρίαν ἔχουσαν []

stretching the bounds of English style one might render thisliterally as

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

You showedwhat ratio the shadow had to the shadow

the pyramid [as] having to the staff

It would be clearer to reverse the order of the last two linesIf the pyramidrsquos height and shadow have lengths P and L theshadow and height being measured from the center of the basewhile the lengths of Thalesrsquos height and shadow are p and ℓthen we may write the claim as

P p L ℓ ()

If not theoretically this must mean practically

P middot ℓ = L middot p ()

that is the rectangle of dimensions P and ℓ is equal to the rect-angle of dimensions L and p Then what is being attributedto Thales is something like the rectangular case of Elementsi

In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another

Thus in the parallelogram ABCD in Figure where AGCis a straight line the parallelograms BG and GD are equalI pause to note that Euclidrsquos two-letter notation for parallel-ograms here is not at all ambiguous in Euclidrsquos Greek wherea diagonal of a parallelogram may be ἡ ΑΒ while the parallel-ogram itself is τὸ ΑΒ the articles ἡ and τό are feminine andneuter respectively Euclidrsquos word παραπλήρωμα for comple-ment is neuter like παραλληλόγραμμον ldquoparallelogramrdquo itselfwhile γραμμή ldquolinerdquo is feminine This observation made in[] is based on similar observations by Reviel Netz []

In Figure the parallelograms BG and GD are equalbecause they are the result of subtracting equal triangles from

Thalesrsquos Theorem

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

A

B C

D

EG

H

F

K

Figure Elements i

equal triangles the equalities of the triangles (AEG = GHAand so forth) are by Elements i The theorem that Plutarchattributes to Thales may now simply be the following

Theorem In equiangular right triangles the rectangles

bounded by alternate legs are equal

Proof Let the equiangular right triangles be ABC and AEFin Figure We shall show

AF middotAB = AE middot AC ()

It will be enough to show AF middot EB = AE middot FC since wecan then add the common rectangle AE middot AF to either sideLet rectangles AG and AH be completed as by the methodwhereby Euclid constructs a square in Elements i this givesus also the rectangle AELF Let BH and CG be extendedto meet at D by Elements i it will be enough to showthat the diagonal AL extended passes through D or in otherwords L lies on the diagonal AD of the rectangle ABDC But

The Nine-point Conic

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

A B

C

E

F

G

HL

A B

C D

E

F

G

HL

Figure The rectangular case of Thalesrsquos Theorem

triangles AFE and FAL are congruent by Elements i andlikewise triangles ACB and CAD are congruent Thus

angFAL = angAFE = angACB = angCAD

which is what we wanted to show

The foregoing is a special case of the following theoremwhich does not require the special case in its proof

Theorem In triangles that share an angle the parallelo-

grams in this angle that are bounded by alternate sides of the

angle are equal if and only if the triangles are equiangular

Proof Let the triangles be ABC and AEF in Figure andlet the parallelograms AG and AH be completed Let the diag-onals CE and BF be drawn If the triangles are equiangularthen we have

FE CB ()

by Elements i In this case as in Euclidrsquos proof of vi thetriangles FEC and EFB are equal by i Adding triangle

Thalesrsquos Theorem

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

A B

C

E

F

G

H

Figure Intermediate form of Thalesrsquos Theorem

AEF in common we obtain the equality of the triangles AECand AFB This gives us the equality of their doubles andthese by i are just the parallelograms AG and AH

Conversely if these parallelograms are equal then so aretheir halves the triangles AEC and AFB hence the trianglesFEC and EFB are equal so () holds by Elements i andthen the original triangles ABC and AEF are equiangular byElements i

Proclus [ ] inadvertently gives evidence that Thalescould use Theorem if not Theorem Discussing Elementsi which is the triangle-congruence theorem whose two partsare now abbreviated as ASA and AAS [ p ] andwhich we used earlier to prove the concurrence of the mediansof a triangle Proclus says

Eudemus in his history of geometry attributes the theoremitself to Thales saying that the method by which he is re-ported to have determined the distance of ships at sea showsthat he must have used it

The Nine-point Conic

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

If Thales really used Euclidrsquos i for measuring distances ofships it may indeed have been by the method that Heathsuggests [ pp ndash] climb a tower note the angle of de-pression of the ship then find an object on land at the sameangle The objectrsquos distance is that of the ship This obviatesany need to know the height of the tower or to know propor-tions Supposedly one of Napoleonrsquos engineers measured thewidth of a river this way

Nonetheless Gow [ p ] observes plausibly that themethod that Heath will propose is not generally practicalThales must have had the more refined method of similar tri-angles

It is hardly credible that in order to ascertain the distanceof the ship the observer should have thought it necessaryto reproduce and measure on land in the horizontal planethe enormous triangle which he constructed in imaginationin a perpendicular plane over the sea Such an undertakingwould have been so inconvenient and wearisome as to de-prive Thalesrsquo discovery of its practical value It is thereforeprobable that Thales knew another geometrical propositionviz lsquothat the sides of equiangular triangles are proportionalrsquo(Euc VI )

But Proposition vi is overkill for measuring distances Allone needs is the case of right triangles in the form of Theorem above This must be the real theorem that Gow goes on todiscuss

And here no doubt we have the real import of those Egyptiancalculations of seqt which Ahmes introduces as exercises inarithmetic The seqt or ratio between the distance of theship and the height of the watch-tower is the same as thatbetween the corresponding sides of any small but similartriangle The discovery therefore attributed to Thales is

Thalesrsquos Theorem

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

probably of Egyptian origin for it is difficult to see whatother use the Egyptians could have made of their seqt whenfound It may nevertheless be true that the proposition EucVI was not known as now stated either to the Egyptiansor to Thales It would have been sufficient for their purposesto know inductively that the seqts of equiangular triangleswere the same

Gow is right that Euclidrsquos Proposition vi need not have beenknown But what he seems to mean is that the Egyptians andThales need only have had a knack for applying the theoremwithout having stepped back to recognize the theorem as suchThis may be so but there is no reason to think they had aknack for applying the theorem in full generality

To establish that theorem Thalesrsquos Theorem in full gener-ality we shall prove that in the proof of Theorem equation() still holds even when applied to Figure in the proofof Theorem To do this we shall rely on the converse of El-

ements i in Figure if the parallelograms BG and GDare equal then the point G must lie on the diagonal AD Wecan prove this by contradiction or by contraposition If G didnot lie on the diagonal then we should be in the situation ofFigure where now parallelograms BN and ND are equalbut BG is part of BN and ND is part of GD so BG is lessthan GD by Euclidrsquos Common Notion

In Euclidrsquos proof of the Pythagorean Theorem Elements

i three auxiliary straight lines concur Heath [ Vol p] passes along Herorsquos proof of this including as a lemmathe converse of i Herorsquos proof is direct but relies on Ele-

ments i equal triangles lying on the same side of the samebase are in the same parallels We used this in proving The-orem and it is the converse of i Euclid proves it bycontradiction using Common Notion

The Nine-point Conic

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

A

B C

D

LN

H

M

K

E FG

Figure Converse of Elements i

Theorem If two parallelograms share an angle the paral-

lelograms are equal if and only if the rectangles bounded by the

same sides are equal

Proof Let the parallelograms be AG and AH in Figure Supposing them equal we prove () namely AF middot AB =AE middot AC Let the parallelogram ABDC be completed Bythe converse of Elements i the point L lies on the diag-onal AD Now erect the perpendiculars AR and AM (usingElements i) and make them equal to AE and AF respec-tively as by drawing circles Each of the parallelograms NEand SF is equal to a rectangle of sides equal to AE and AF by Elements i Therefore A lies on the diagonal LX againby the converse of Elements i so A and L both lie on thediagonal DX of the large parallelogram Consequently theparallelograms SC and NB are equal but they are also equalto AE middot AC and AF middot AB respectively The converse is simi-lar

Thalesrsquos Theorem

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

AB

C D

E

F

G

HL

MN P Q

R

S

T

U

X

Figure Thalesrsquos Theorem

Theorems and together are what we are calling ThalesrsquosTheorem provided we can establish Elements vi

If four straight lines be proportional the rectangle containedby the extremes is equal to the rectangle contained by themeans and [conversely]

To do this we need a proper theoretical definition of propor-tion

Thales and Desargues

I suggested that the equivalence of the proportion () withthe equation () was ldquopracticalrdquo To read the colons in theexpression

a b c d ()

we can say any of the followingbull a b c and d are proportional

The Nine-point Conic

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

bull a is to b as c is to dbull the ratio of a to b is the same as the ratio of c to d

From the last clause we can delete the phrase ldquothe same asrdquothis is effectively what Plutarch does in the passage quotedabove although the translator puts the phrase back in Theequation

a middot d = b middot c ()

does not in itself express a property of the ordered pair (a b)that is the same as the corresponding property of (c d) itexpresses only a relation between the pairs Immediately wehave a middot b = b middot a and if () holds so does c middot b = d middot a so therelation expressed by () between (a b) and (c d) is reflexiveand symmetric The relation is not obviously transitive if() holds and c middot f = d middot e it is not obvious that a middot f = b middot eIt would be true for example if we allowed passage to a thirddimension obtaining from the hypotheses

a middot d middot f = b middot c middot f b middot c middot f = b middot d middot e

whence presumably the desired conclusion would follow butthis would be a theorem Therefore () alone cannot con-stitute the definition of () I have argued this elsewhere inthe context of Euclidrsquos number theory []

Suppose now that each of the letters in () stands for alength not a number but the class of Euclidean boundedstraight lines that are equal to a particular straight line Ipropose to define () to mean that for all lengths x and y

b middot x = a middot y lArrrArr d middot x = c middot y ()

In other words () means that the sets

(x y) b middot x = a middot y (x y) d middot x = c middot y

Thales and Desargues

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

are the same This definition ensures logically that the re-lation of having the same ratio is transitive as any relationdescribed as a sameness should be The definition also en-sures that () implies () let (x y) = (c d) in () Thedefinition avoids the ldquoArchimedeanrdquo assumption required bythe definition attributed to Eudoxus found in Book v of theElements However we need to prove that () implies ()This implication is Elements vi for the new definition ofproportion

Theorem If of four lengths the rectangle bounded by the

extremes is equal to the rectangle bounded by the means then

the lengths are in proportion

Proof Supposing we are given four lengths a b c d such that() holds we want to show () as defined by () It isenough to show that if two lengths e and f are such that

a middot f = b middot e ()

thenc middot f = d middot e ()

In Figure let AH and AL have lengths c and e respectivelyDraw AG parallel to HL (Elements i) and let AG havelength a Complete the parallelogram ABDC so that GB andHC have lengths b and d respectively By () and Theorem the parallelograms GP and HN are equal Now complete theparallelogram ABFE and denote by g the length of LE Theparallelograms GR and GP are equal by Elements i BothLM and HK have length a (Elements i) so parallelogramsLQ and HN are equal by Elements i Thus GR and LQare equal Hence by Theorem we have

a middot g = b middot e

The Nine-point Conic

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

bA

bB

b

C

b Db

E

b

F

bG

bH

bK

bL

b

M

a b

c

d

e

g

b

N

b P

b

Q

b

R

Figure Transitivity

so a middot g = a middot f by () and therefore g = f by CommonNotion Since LH EC we have () by Theorem

There are easy consequences corresponding to Propositions and of Elements Book v but these propositions are notthemselves so easy to prove with the Eudoxan definition ofproportion

Corollary (Alternation)

a b c d =rArr a c b d

Proof Each proportion is equivalent to a middot d = b middot c

In a note on v Heath [ Vol pp ndash] observesthat the proposition is easier to prove whenmdashas for usmdashthemagnitudes being considered are lengths He quotes the text-book of Smith and Bryant [ pp ndash] which derives thespecial case from vi parallelograms and triangles under the

Thales and Desargues

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

same height are to one another as their bases We might writethis as

a b a middot c b middot cOne could take this as a definition of proportion but then onehas the problem of transitivity as before If

a b c middot e d middot e

for some e meaning c middot e = a middot f and d middot e = b middot f for some f one has to show the same for arbitrary e

Corollary (Cancellation)

a b d e

b c e f

=rArr a c d f

Proof Under the hypothesis by alternation a d b e andb e c f Then a d c f since sameness of ratio istransitive by definition

Theorems and together constitute Thalesrsquos TheoremIn proving Theorem we effectively showed in Figure

BC GH amp CE HL =rArr BE GL ()

provided alsoCE AB ()

Then by Thalesrsquos Theorem itself since sameness of ratio istransitive () holds without need for () This is De-sarguesrsquos Theorem if the straight lines through correspondingvertices of two triangles concur and two pairs of correspond-ing sides of the triangles are parallel then the third pair mustbe parallel as in Figure We have proved this from Book

The Nine-point Conic

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

b

b

b

b

b

b

b

Figure Desarguesrsquos Theorem

i of Euclid without the Archimedean assumption of Book vthough with Common Notion for areas

Desarguesrsquos Theorem has other cases in the Euclidean planeWhen we add to this plane the ldquoline at infinityrdquo thus obtain-ing the projective plane then the three pairs of parallel linesin the theorem intersect on the new line But then any lineof the projective plane can serve as a line at infinity addedto a Euclidean plane A way to show this is by the kind ofcoordinatization that we are in the process of developing

Locus problems

Fixing a unit length Descartes [ p ] defines the productab of lengths a and b as another length given by the rule thatwe may express as

1 a b ab

Denoting Decartesrsquos product thus by juxtaposition alonewhile continuing to denote with a dot the area of a rectan-

Locus problems

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

gle with given dimensions by Theorem we have

ab middot 1 = a middot b

In particular Cartesian multiplication is commutative and itdistributes over addition since

a middot b = b middot a a middot (b+ c) = a middot b+ a middot c

and from Common Notion we have

d middot 1 = e middot 1 =rArr d = e

That Cartesian multiplication is associative can be seen fromthe related operation of composition of ratios given by therule

(a b) amp (b c) a c

One may prefer to use the sign = of equality here rather thanthe sign of sameness of ratio if one judges it not to beimmediate that the compound ratio (a b) amp (b c) dependsnot merely on the ratios a b and b c but on their givenrepresentations in terms of a b and c Nonetheless it doesdepend only on the ratios by Corollary Then compositionof ratios is immediately associative We have generally

(a b) amp (c d) e d

provided a b e c and such e can be found by the methodof Elements i and Then

(a 1) amp (b 1) ab 1

and from this we can derive a(bc) = (ab)c Also

(a 1) amp (1 a) 1 1

The Nine-point Conic

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

so multiplication is invertibleOne can define the sum of two ratios as one defines the sum

of fractions in school by finding a common denominator Inparticular

(a 1) + (b 1) = (a+ b) 1

where now it does seem appropriate to start using the signof equality Now both lengths and ratios compose fields infact ordered fields which are isomorphic under x 7rarr x 1Descartes shows this implicitly in order to solve ancient prob-lems One may object that we have not introduced additiveinverses whether of lengths or of ratios We can do this byassigning to each class of parallel straight lines a directionso that the signed length of BA is the additive inverse of thesigned length of AB

Descartes [ p n ] alludes to a passage in the Col-

lection where Pappus [ pp ndash] describes three kindsof geometry problem plane as being solved by means ofstraight lines and circles which lie in a plane solid as requir-ing also the use of conic sections which are sections of a solidfigure the cone and linear as involving more complicatedlines that is curves An example of a linear problem thenwould be the quadrature or squaring of the circle achieved bymeans of the quadratrix or ldquotetragonizerrdquo (τετραγωνίζουσα)which Pappus [ pp ndash] defines as being traced in asquare such as ABGD in Figure by the intersection oftwo straight lines one horizontal and moving from the topedge BG to the bottom edge AD the other swinging aboutthe lower left corner A from the left edge AB to the bottomedge AD If there is a point H where the quadratrix meetsthe lower edge of the square then as Pappus shows

BD AB AB AH

Locus problems

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

x

y

θ

A

B G

DH

Figure The quadratrix

where BD is the circular arc centered at A In modern termswith variables as in the figure AB being taken as a unit

θ

y=

π

2 tan θ =

y

x

so

x =2

π

middot θ

tan θ

As θ vanishes x goes to 2π This then is the length ofAH Pappus points out that we have no way to constructthe quadratrix without knowing where the point H is in thefirst place He attributes this criticism to one Sporus aboutwhom we apparently have no source but Pappus himself [p n ]

A solid problem that Pappus describes [ pp ndash] isthe four-line locus problem find the locus of points suchthat the rectangle whose dimensions are the distances to two

The Nine-point Conic

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

given straight lines bears a given ratio to the rectangle whosedimensions are the distances to two more given straight linesAccording to Pappus theorems of Apollonius were needed tosolve this problem but it is not clear whether Pappus thinksApollonius actually did work out a full solution By the lastthree propositions namely ndash of Book III of the Conics ofApollonius it is implied that the conic sections are three-lineloci that is solutions to the four-line locus problem when twoof the lines are identical Taliaferro [ pp ndash] works outthe details and derives the theorem that the conic sections arefour-line loci

Descartes works out a full solution to the four-line locusproblem [ pp ndash] He also solves a particular five-linelocus problem where four of the straight linesmdashsay ℓ0 ℓ1ℓ2 and ℓ3mdashare parallel to one another each a distance a fromthe previous while the fifth linemdashℓ4mdashis perpendicular to them[ pp ndash] What is the locus of points such that the prod-uct of their distances to ℓ0 ℓ1 and ℓ3 is equal to the productof a with the distances to ℓ2 and ℓ4 One can write down anequation for the locus and Descartes does Letting distancesfrom ℓ4 and ℓ2 be x and y respectively Descartes obtains

y3 minus 2ay2 minus a2y + 2a3 = axy ()

This may allow us to plot points on the desired locus obtainingthe bold solid curve in Figure but we could already dothat The equation is thus not a solution to the locus problemsince it does not tell us what the locus is But Descartesshows that the locus is traced by the intersection of a movingparabola with a straight line passing through one fixed pointand one point that moves with the parabola as suggested bythe dashed lines in Figure The parabola has axis slidingalong ℓ2 and the latus rectum of the parabola is a so the

Locus problems

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

ℓ0 ℓ1 ℓ2 ℓ3

ℓ4 bc

bc

Figure Solution of a five-line locus problem

The Nine-point Conic

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

parabola is given by ax = y2 when its vertex is on ℓ4 Thestraight line passes through the intersection of ℓ0 and ℓ4 andthrough the point on the axis of the parabola whose distancefrom the vertex is a

We shall be looking at latera recta again later meanwhileone may consult my article ldquoAbscissas and Ordinatesrdquo [] tolearn more than one ever imagined wanting to know about theterminology

Descartesrsquos solution of a five-line locus problem is apparentlyone that Pappus would recognize as such Thus Descartesrsquosalgebraic methods would seem to represent an advance andnot just a different way of doing mathematics As Descartesknows [ p n ] Pappus [ pp -] could formulate

the 2n- and (2n + 1)-line locus problems for arbitrary n Ifn gt 3 the ratio of the product of n segments with the productof n segments can be understood as the ratio compounded ofthe respective ratios of segment to segment Given 2n lengthsa1 an b1 bn we can understand the ratio of theproduct of the ak to the product of the bk as the compositeratio

(a1 b1) amp middot middot middotamp (an bn)

Pappus recognizes this Descartes expresses the solution of the2n-line locus problem as an nth-degree polynomial equation inx and y where y is the distance from a point of the locus toone of the given straight lines and x is the distance from agiven point on that line to the foot of the perpendicular fromthe point of the locus Today we call the line the x-axis andthe perpendicular through the given point on it the y-axis

but Descartes does not seem to have done this expressly

Descartes does effectively allow oblique axes The originallocus problems literally involve not distances to the given lines

Locus problems

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

but lengths of straight lines drawn at given angles to the givenlines

The nine-point conic

The nine-point conic is the solution of a locus problem Thesolution had been known earlier but apparently the solutionwas first remarked on in by Maxime Bocirccher [] whosays

It does not seem to have been noticed that a few well-known facts when properly stated yield the following directgeneralization of the famous nine point circle theoremmdash

Given a triangle ABC and a point P in its plane a coniccan be drawn through the following nine points

() The middle points of the sides of the triangle

() The middle points of the lines joining P to the verticesof the triangle

() The points where these last named lines cut the sidesof the triangle

The conic possessing these properties is simply the locusof the centre of the conics passing through the four pointsA B C P (cf Salmonrsquos Conic Sections p Ex andp Ex )

Bocirccherrsquos references fit the sixth and tenth editions of SalmonrsquosTreatise on Conic Sections dated and respectively[ ] but in the ldquoThird Edition revised and enlargedrdquo dated [] the references would be p Ex and p Art

Following Salmon we start with a general equation

ax2 + 2hxy + by2 + 2gx+ 2fy + c = 0 ()

The Nine-point Conic

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

of the second degree We do not assume that x and y aremeasured at right angles to one another we allow obliqueaxes We do assume ab 6= 0 If we wish we can eliminatethe xy term by redrawing the x-axis and changing its scaleso that the point now designated by (x y) is the one that wascalled (x + hya y) before The curve that was defined by() is now defined by

ax2 +

(

bminus h2

a

)

y2 + 2gx+ 2

(

f minus gh

a

)

y + c = 0 ()

If the linear terms in () are absent then they are absentfrom () as well

The equation () defines an ellipse or hyperbola no mat-ter what the angle is between the axes or what their relativescales are This is perhaps not strictly speaking high-schoolknowledge One may know from school that an equation

x2

a2plusmn y2

b2= 1

defines a certain curve called ellipse or hyperbola dependingon whether the upper or lower sign is taken But one assumesthat x and y are measured orthogonally One does not learnwhy the curves are named as they are [] and one does notlearn that if the appropriate oblique axes are chosen then thecurve has an equation of the same form But this is just whatBook I of the Conics of Apollonius is devoted to showing []

Going back to the original axes and the curve defined by() we translate the axes so that the new origin is thepoint formerly called (xprime yprime) The curve is now given by

ax2 + 2hxy + by2 + 2gprimex+ 2f primey + cprime = 0

The nine-point conic

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

where

gprime = axprime + hyprime + g f prime = byprime + hxprime + f ()

and we are not interested in cprime The curve given by () hascenter at (xprime yprime) just in case (gprime f prime) = (0 0)

For a complete quadrangle in which one pair (at least) ofopposite sides are not parallel those sides determine a coor-dinate system In this system suppose the vertices of thecomplete quadrilateral are (λ 0) and (λprime 0) on the x-axis and(0 micro) and (0 microprime) on the y-axis Let the conic given by ()pass through these four points Setting y = 0 we obtain theequation

ax2 + 2gx+ c = 0

which must have roots λ and λprime so that the equation is

a(

x2 minus (λ+ λprime)x+ λλprime)

= 0

From this we obtain

2g = minusa(λ+ λprime) c = aλλprime

Likewise setting x = 0 in () yields the equation

by2 + 2fy + c = 0

which must be

b(

y2 minus (micro+ microprime)y + micromicroprime)

= 0

from which we obtain

2f = minusb(micro + microprime) c = bmicromicroprime

The Nine-point Conic

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

From the two expressions for c we have

aλλprime = bmicromicroprime

In () we are free to let a = micromicroprime Then b = λλprime and ()becomes

micromicroprimex2 + 2hxy + λλprimey2

minus micromicroprime(λ+ λ)xminus λλprime(micro+ microprime)y + λλprimemicromicroprime = 0 ()

By the computations for () the center of the conic in ()satisfies

2micromicroprimex+ 2hy minus micromicroprime(λ+ λprime) = 0

2λλprimey + 2hxminus λλprime(micro+ microprime) = 0

Eliminating h yields

2micromicroprimex2 minus micromicroprime(λ+ λprime)x = 2λλprimey2 minus λλprime(micro+ microprime)y ()

This is the equation of an ellipse or hyperbola passing throughthe origin We can also write the equation as

micromicroprimex

(

xminus λ+ λprime

2

)

= λλprimey

(

y minus micro+ microprime

2

)

()

this shows that the conic passes through the midpoints ofthe sides of the complete quadrangle that lie along the axesBy symmetry the conic passes through the midpoints of allsix sides of the quadrangle and through its three diagonalpointsmdashif these exist that is if none of the three pairs ofopposite sides of the quadrangle are parallel

I suggested earlier that for Descartes to solve a five-line locusproblem it was not enough to find the equation () he had

The nine-point conic

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

to describe the solution geometrically We do have a thoroughgeometric understanding of solutions of second-degree equa-tions like () and () or even () Alternatively thereis a modern ldquosyntheticrdquo approach [ p ] that is anapproach based not on field axioms but on geometric axiomshere axioms for a projective plane but with a line at infinitydesignated so that midpoints of segments of other lines canbe defined

I do not know how directly the nine-point conic can be de-rived from the work of Apollonius Here I shall just want tolook briefly at an example of how Apollonius uses areas in away not easily made algebraic David Hilbert shows how alge-bra is possible without a priori assumptions about areas butit is not clear how much is gained

The Nine-point Conic

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Lengths and Areas

Algebra

The points of an unbounded straight line are elements of anordered abelian group with respect to the obvious notion of ad-dition once an origin and a direction have been selected If weset up two straight lines at right angles to one another lettingtheir intersection point be the origin then after fixing also aunit length we obtain Hilbertrsquos definition of multiplication asin Figure the two oblique lines being parallel

Not having accepted the Eudoxan definition of proportionor Thalesrsquos Theorem Hilbert can still show that multiplicationis commutative and associative He does this by means of whathe calls Pascalrsquos Theorem although it was referred to in theIntroduction above as Pappusrsquos Theorem if the vertices of a

1 b

a

ab

0

Figure Hilbertrsquos multiplication

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

b b

b

b

b

b

b

0a db

d(ca)

b

ca

c(db)

Figure Pappusrsquos Theorem

hexagon lie alternately on two straight lines in the projectiveplane then the intersection points of the three pairs of oppositesides lie on a straight line Pascal announced the generalizationin which the original two straight lines can be an arbitraryconic section [ ]

When we give Pappusrsquos Theorem in the Euclidean planethe simplest case occurs as in Figure labelled for prov-ing commutativity and associativity of multiplication (in casethe angle at 0 is right) In general if there are two pairs ofparallel opposite sides of the hexagon that is woven like a spi-derrsquos web or catrsquos cradle across the angle then the third pairof opposite sides are parallel as well By the numbering inHultschrsquos edition of Pappusrsquos Collection [] which is appar-ently the numbering made originally by Commandinus [ ppndash ] the result is Proposition in Book vii it is alsonumber viii of Pappusrsquos lemmas for the now-lost Porisms ofEuclid

Lemma viii seems to have been sadly forgotten The caseof Pappusrsquos Theorem where the three pairs of opposite sides

Lengths and Areas

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

a

c

α

Figure Hilbertrsquos trigonometry

all intersect is Propositions and or Lemmas xii andxiii these cover the cases when the straight lines on which thevertices of the hexagon lie are parallel and not respectivelyKline cites only Proposition as giving Pappusrsquos Theorem[ p ] In his summary of most of Pappusrsquos lemmas forEuclidrsquos Porisms Heath [ p ndash] lists Propositions and as constituting Pappusrsquos Theorem The lasttwo are converses of the first two as Pappus states them Itis not clear that Pappus recognizes a single theme behind theseveral of his lemmas that constitute the theorem named forhim He omits the case where exactly one pair of oppositesides are parallel

Heath omits to mention Proposition at all This is astrange oversight for an important theorem Pappus proves itby means of areas using Euclidrsquos i as we proved Theorem Without using areas Hilbert gives two elaborate proofsone of which in the situation of Figure uses the notation

a = αc

for what today might be written as a = c cosαFor proving associativity and commutativity of multiplica-

tion Hartshorne has a more streamlined approach in Geom-

Algebra

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

AB C

D

E

c

ac

bc

b(ac)

a(bc)

Figure Associativity and commutativity

etry Euclid and Beyond [ pp ndash] In Figure AChas the two lengths indicated so these are the same that is

a(bc) = b(ac) ()

Letting c = 1 gives commutativity then this with () givesassociativity Distributivity follows from Figure where

ab+ ac = a(b+ c)

The advantage of defining multiplication in terms of lengthsalone (and right angles and parallelism but not parallelo-grams or other bounded regions of the plane) is that it allowsall straight-sided regions to be linearly ordered by size with-out assuming a priori that the whole region is greater thanthe part

Lengths and Areas

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

b

ab

c

ac

Figure Distributivity

Euclid provides for an ordering in Elements i and which show that every straight-sided region is equal to a rect-angle on a given base Finding the rectangle involves iShowing the rectangle unique requires the converse of iwhich in turn requires Common Notion for areas The trian-gle may be equal to the rectangle on the same base with halfthe height but there are three choices of base and so threerectangles result that are equal to the triangle When theyare all made equal to rectangles on the same base why shouldthey have the same height Common Notion is one reasonbut Hilbert doesnrsquot need it

The triangle is equal to the rectangle whose base is half theperimeter of the triangle and whose height is the radius ofthe inscribed circle see Figure where the triangle ABCis equal to the sum of AG middot GD BG middot GD and CE middot EDbut ED = GD However if ABC is cut into two trianglesand rectangles equal to the two triangles are found addedtogether and made equal to a rectangle whose base is againhalf the perimeter of ABC we need to know that the heightis equal to GD

Algebra

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

AB

C

D

E

F

G

Figure Circle inscribed in triangle

AB

C

h

k

c

b

α

Figure Area of triangle in two ways

In the notation of Figure we have

h = c sinα k = b sinα

where sinα stands for the appropriate length and the multi-plication is Hilbertrsquos and then

bh = bc sinα = cb sinα = ck

Thus we can define the area of ABC unambiguously as thelength bh2 With this definition when the triangle is divided

Lengths and Areas

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

into two parts or indeed into many parts as Hilbert showsthe area of the whole is the sum of the areas of the parts

Moreover if two areas become equal when equal areas areadded then the two original areas are themselves equal thisis Euclidrsquos Common Notion but Hilbert makes it a theoremOne might classify this theorem with the one that almost ev-ery human being learns in childhood but almost nobody everrecognizes as a theorem no matter how you count a finite setyou always get the same number It can be valuable to be-come clear about the basics as I have argued concerning thelittle-recognized distinction between induction and recursion[]

Apollonius

We look at a proof by Apollonius in order to considerDescartesrsquos idea quoted in the Introduction that ancientmathematicians had a secret method

We first set the stage this is done in more detail in [] Acone is determined by a base which is a circle and a vertex

not in the plane of the base but not necessarily hovering rightover the center of the base either the cone may be obliqueThe surface of the cone is traced out by the straight lines thatpass through the vertex and the circumference of the base Adiameter of the base is also the base of an axial triangle

whose apex is the vertex of the cone If a chord of the baseof the cone cuts the base of the axial triangle at right anglesthen a plane containing the chord and parallel to a side ofthe axial triangle cuts the surface of the cone in a parabola

The cutting plane cuts the axial triangle in a straight line thatis called a diameter of the parabola because the line bisects

Apollonius

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

the chords of the parabola that are parallel to the base of thecone Half of such a chord is an ordinate it cuts off from thediameter the corresponding abscissa the other endpoint ofthis being the vertex of the parabola There is some boundedstraight line the latus rectum such that when the squareon any ordinate is made equal to a rectangle on the abscissathe other side of the rectangle is precisely the latus rectumthis is Proposition i of Apollonius and it is the reason forthe term parabola meaning application

The tangent to the parabola at the vertex is parallel to theordinates We are going to show that every straight line paral-lel to the diameter is another diameter with a correspondinglatus rectum and latera recta are to one another as the squareson the straight lines each of which is drawn drawn tangent tothe parabola from vertex to other diameter

In Cartesian terms we may start with a diameter that is anaxis in the sense of being at right angles to its ordinates Ifthe latus rectum is ℓ then the parabola can be given by

ℓy = x2 ()

As in Figure the tangent to the parabola at (a a2ℓ) cutsthe y-axis at minusa2ℓ this can be shown with calculus but it isProposition i of Apollonius The tangent then is

ℓy = 2axminus a2

We shall take this and x = a as new axes say xprime- and yprime-axesIf d is the distance between the new origin and the intersectionof the xprime-axis with the y-axis then since the x- and y-axes areorthogonal by the Pythagorean Theorem (Elements i) wehave

d2 = a2 +

(

2a2

ℓ

)2

=a2

ℓ2(ℓ2 + 4a2)

Lengths and Areas

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

ℓ

ℓ

a

a2ℓ

minusa2ℓ

minus2a

y

x

yprime

xprime

Figure Change of coordinates

Apollonius

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Let b =radicℓ2 + 4a2 so d = baℓ Then

ℓxprime =ℓd

a(xminus a) = bxminus ba ℓyprime = ℓy minus 2ax+ a2

so

bx = ℓxprime + ba ℓy = ℓyprime +2a

b(ℓxprime + ba)minus a2

= ℓyprime +2ℓa

bxprime + a2

Plugging into () yields

ℓyprime +2ℓa

bxprime + a2 =

(

ℓ

bxprime + a

)2

b2yprime = ℓ(xprime)2

In particular the new latus rectum is b2ℓ which is as claimedsince b2ℓ2 = d2a2

For his own proof of this Apollonius uses a lemma Propo-sition i [ p ndash] In Figure we have a parabolaΓ∆Β with diameter ΒΘ Here ∆Ζ and ΓΘ are ordinates and ΒΗ

is parallel to these so it is tangent to the parabola at Β Thestraight line ΑΓ is tangent to the parabola at Γ which meansby i

ΑΒ = ΒΘ ()

ThenΑΓΘ = ΗΘ ()

the latter being the parallelogram with those opposite verticesThe straight line ∆Ε is drawn parallel to ΓΑ Then trianglesΑΓΘ and Ε∆Ζ are similar so their ratio is that of the squares ontheir bases those bases being the ordinates mentioned Then

Lengths and Areas

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Γ

Α Ε

∆

Β

Η

Ζ Θ

Figure Proposition i of Apollonius

the abscissas ΒΘ and ΒΖ are in that ratio and hence the par-allelograms ΗΘ and ΗΖ are in that ratio By () then ΑΓΘ

has the same ratio to ΗΖ that it does to Ε∆Ζ Therefore

Ε∆Ζ = ΗΖ

The relative positions of ∆ and Γ on the parabola are irrelevantto the argument this will matter for the next theorem

In Figure now Κ∆Β is a parabola with diameter ΒΜand Γ∆ is tangent to the parabola and through ∆ parallelto the diameter straight line ∆Ν is drawn and extended toΖ so that ΖΒ is parallel to the ordinate ∆Ξ A length Η (notdepicted) is taken such that

Ε∆ ∆Ζ Η 2Γ∆ ()

Through a random point Κ on the parabola ΚΛ is drawnparallel to the tangent Γ∆ We shall show

ΚΛ2 = Η middot ∆Λ ()

Apollonius

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Ξ

∆

Γ Μ

Κ

Π

ΝΖ

Β

Λ

Ε

Figure Proposition i of Apollonius

so that ∆Ν serves as a new diameter of the parabola with cor-responding latus rectum Η This is Proposition i of Apol-lonius

Since as before ΓΒ = ΒΞ we have

ΕΖ∆ = ΕΓΒ ()

Let ordinate ΚΝΜ be drawn Adding to either side of ()the pentagon ∆ΕΒΜΝ we have

ΖΜ = ∆ΓΜΝ

Let ΚΛ be extended to Π By the lemma that we proved aboveΚΠΜ = ΖΜ Thus

ΚΠΜ = ∆ΓΜΝ

Lengths and Areas

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Subtracting the trapezoid ΛΠΜΝ gives

ΚΛΝ = ΛΓ

From this by Theorem above we have

ΚΛ middot ΛΝ = 2Λ∆ middot ∆Γ ()

We now compute

ΚΛ2 ΚΛ middot ΛΝ ΚΛ ΛΝ

Ε∆ ∆Ζ

Η 2Γ∆ [by ()]

Η middot Λ∆ 2Λ∆ middot Γ∆

Η middot Λ∆ ΚΛ middot ΛΝ [by ()]

which yields () We have assumed Κ to be on the other sideof ∆Ν from ΒΜ The argument can be adapted to the othercase Then as a corollary we have that ∆Ν bisects all chordsparallel to ∆Γ In fact Apollonius proves this independentlyin Proposition i

Could Apollonius have created the proof of i for pedagog-ical or ideological reasons after verifying the theorem itself byCartesian methods such we we employed I have not foundany reason to think so Before Apollonius it seems that onlythe right cone was studied and the only sections consideredwere made by planes that were orthogonal to straight linesin the surface of the cone [ p xxiv] Whether an ellipseparabola or hyperbola was obtained depended on the angleat the vertex of the cone The recognition that the cone canbe oblique and every section can be obtained from every coneseems to be due to Apollonius That our Cartesian argument

Apollonius

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

started with orthogonal axes corresponds to starting with aright cone On the other hand this feature was not essentialto the argument we did not really need the parameter b

Lengths and Areas

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Unity

In addition to Elements i which is ASA and AAS asdiscussed earlier Proclus [ ] attributesto Thales three more of Euclidrsquos propositions depicted in Fig-ure

) the diameter bisects the circle as in the remark on oraddendum to the definition of diameter given at thehead of the Elements

) the base angles of an isosceles triangle are equal (i)) vertical angles are equal (i)

Kant alludes to the second of these theorems in the Prefaceto the B Edition of the Critique of Pure Reason in a purplepassage of praise for the person who discovered the theorem[ b xndashxi p ndash]

Mathematics has from the earliest times to which the his-tory of human reason reaches in that admirable people the

Figure Symmetries

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Greeks traveled the secure path of a science Yet it must notbe thought that it was as easy for it as for logicmdashin whichreason has to do only with itselfmdashto find that royal path orrather itself to open it up rather I believe that mathemat-ics was left groping about for a long time (chiefly among theEgyptians) and that its transformation is to be ascribed toa revolution brought about by the happy inspiration of asingle man in an attempt from which the road to be takenonward could no longer be missed and the secure course of ascience was entered on and prescribed for all time and to aninfinite extent The history of this revolution in the way ofthinkingmdashwhich was far more important than the discoveryof the way around the famous Capemdashand of the lucky onewho brought it about has not been preserved for us But thelegend handed down to us by Diogenes Laertiusmdashwho namesthe reputed inventor of the smallest elements of geometri-cal demonstration even of those that according to commonjudgment stand in no need of proofmdashproves that the mem-ory of the alteration wrought by the discovery of this newpath in its earliest footsteps must have seemed exceedinglyimportant to mathematicians and was thereby rendered un-forgettable A new light broke upon the person who demon-strated the isosceles triangle (whether he was called ldquoThalesrdquoor had some other name)

The boldface is Kantrsquos The editors cite a letter in which Kantconfirms the allusion to Elements i

In apparent disagreement with Kant I would suggest thatrevolutions in thought need not persist they must be madeafresh by each new thinker The student need not realize heror his potential for new thought no matter how open the royalpath may seem to the teacher

Kant refers to the discovery of the southern route aroundAfrica does he mean the discovery by Bartolomeu Dias in

Unity

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

or the discovery by the Phoenicians sailing the otherdirections two thousand years earlier described by Herodotus[ iv pp ndash] The account of Herodotus is madeplausible by his disbelief that in sailing west around the Capeof Good Hope the Phoenicians could have found the sun ontheir right Their route was not maintained which is why Diascan be hailed as its discoverer

Likewise may routes to mathematical understanding not bemaintained Much of ancient mathematics has been lost to usin the slow catastrophe alluded to by the title of The Forgot-

ten Revolution [ p ] Here Lucio Russo points out thatthe last of the eight books of Apollonius on conic sections nolonger exists while Books vndashvii survive only in Arabic trans-lation and we have only Books indashiv in the original GreekPresumably this is because the later books of Apollonius werefound too difficult by anybody who could afford to have copiesmade

We may be able to recover the achievement of Thales ifThales be his name There is no reason in principle why wecannot understand him as well as we understand anybody buttime and loss present great obstacles Kant continues with hisown interpretation of Thalesrsquos thought

For he found that what he had to do was not to trace whathe saw in this figure or even trace its mere concept andread off as it were from the properties of the figure butrather that he had to produce the latter from what he him-self thought into the object and presented (through construc-tion) according to a priori concepts and that in order toknow something securely a priori he had to ascribe to thething nothing except what followed necessarily from what hehimself had put into it in accordance with its concept

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Kant is right if he means that the equality of the base angles ofan isosceles triangle does not follow merely from a figure of anisosceles triangle The figure itself represents only one instanceof the general claim One has to recognize that the figure isconstructed according to a principle which in this case canbe understood as symmetry The three shapes in Figure embody a symmetry that justifies the corresponding theoremseven though Euclid does not appeal to this symmetry in hisproofs

As we discussed Thales is also thought to have discoveredthat the angle in a semicircle is right and he may have donethis by recognizing that any two diameters of a circle are diag-onals of an equiangular quadrilateral Such quadrilaterals arerectangles but this is not so fundamental an observation asthe equality of the base angles of an isosceles triangle in factthe ldquoobservationrdquo can be disputed as it was by Lobachevski[]

Thales is held to be the founder of the Ionian school ofphilosophy In Before Philosophy [ p ] the Frankfortssay of the Ionian school that its members

proceeded with preposterous boldness on an entirely un-proved assumption They held that the universe is an intel-ligible whole In other words they presumed that a singleorder underlies the chaos of our perceptions and further-more that we are able to comprehend that order

For Thales the order of the world was apparently to be ex-plained through the medium of water However this informa-tion is third-hand at best Aristotle says in De Caelo [ IIpp ]

By these considerations some have been led to assert thatthe earth below us is infinite saying with Xenophanes ofColophon that it has lsquopushed its roots to infinityrsquomdashin or-

Unity

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

der to save the trouble of seeking for the cause Otherssay the earth rests upon water This indeed is the oldesttheory that has been preserved and is attributed to Thalesof Miletus It was supposed to stay still because it floatedlike wood and other similar substances which are so consti-tuted as to rest upon water but not upon air As if the sameaccount had not to be given of the water which carries theearth as of the earth itself

Aristotle has only second-hand information on Thales whoseems not to have written any books It may be that Aristo-tle does not understand what questions Thales was trying toanswer Aristotle has his own questions and criticizes Thalesfor not answering them

Aristotle may do a little better by Thales in the Metaphysics[ a pp ndash] where he says

Of the first philosophers then most thought the principleswhich were of the nature of matter were the only principlesof all things Yet they do not all agree as to the numberand the nature of these principles Thales the founder ofthis type of philosophy says the principle is water (for whichreason he declared that the earth rests on water) getting thenotion perhaps from seeing that the nutriment of all thingsis moist and that heat itself is generated from the moist andkept alive by it (and that from which they come to be is aprinciple of all things) He got his notion from this fact andfrom the fact that the seeds [τὰ σπέρματα] of all things havea moist nature and that water is the origin of the nature ofmoist things

In The Idea of Nature [ pp ndash] Collingwood has a poeticinterpretation of this

The point to be noticed here is not what Aristotle says butwhat it presupposes namely that Thales conceived the world

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

of nature as an organism in fact as an animal he maypossibly have conceived the earth as grazing so to speak onthe water in which it floats thus repairing its own tissues andthe tissues of everything in it by taking in water from thisocean and transforming it by processes akin to respirationand digestion into the various parts of its own body Thisanimal lived in the medium out of which it was made as acow lives in a meadow But now the question arose Howdid the cow get there The world was not born it wasmade made by the only maker that dare frame its fearfulsymmetry God

Collingwood is presumably alluding to the sixth and finalstanza of William Blakersquos poem ldquoThe Tygerrdquo []

Tyger Tyger burning brightIn the forests of the nightWhat immortal hand or eyeDare frame thy fearful symmetry

Like most animals the tiger exhibits bilateral symmetry thekind of symmetry shown in Figure though this may not bewhat Blake is referring to his illustration for the poem showsa tiger from the side not the front For Euclid συμμετρία iswhat we now call commensurability symmetry may also bebalance and harmony in a non-mathematical sense [] Butthis would seem to be what Thales saw or sought in theworld and it is akin to his recognition of the unity underlyingall isosceles triangles a unity whereby the equality of the baseangles of each of them can be established once for all Whatmade Thales a philosopher made him a mathematician

Unity

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

Bibliography

[] Apollonius of Perga Apollonius of Perga Treatise on Conic Sec-

tions University Press Cambridge UK Edited by T LHeath in modern notation with introductions including an essay onthe earlier history of the subject

[] Apollonius of Perga Apollonii Pergaei Qvae Graece Exstant Cvm

Commentariis Antiqvis volume I Teubner Edited with Latininterpretation by I L Heiberg First published

[] Apollonius of Perga Conics Books IndashIII Green Lion Press SantaFe NM revised edition Translated and with a note and an ap-pendix by R Catesby Taliaferro with a preface by Dana Densmoreand William H Donahue an introduction by Harvey Flaumenhaftand diagrams by Donahue edited by Densmore

[] Archimedes Archimedis Opera Omnia Cum Commentarius Eu-

tochii volume II B G Teubner Recension of the FlorentineCodex with Latin translation and notes by J L Heiberg

[] Archimedes The Works of Archimedes University Press Cam-bridge UK Edited in Modern Notation With IntroductoryChapters by T L Heath

[] Archimedes The Two Books On the Sphere and the Cylinder vol-ume I of The Works of Archimedes Cambridge University PressCambridge Translated into English together with Euto-ciusrsquo commentaries with commentary and critical edition of thediagrams by Reviel Netz

[] Aristotle The Basic Works of Aristotle Random House New York Edited and with an Introduction by Richard McKeon

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

[] Aristotle De caelo In The Basic Works of Aristotle [] pages ndash Translated by J L Stocks Available at classicsmitedu

Aristotleheavenshtml

[] Aristotle Metaphysica In The Basic Works of Aristotle [] pagesndash Translated by W D Ross Available at classicsmit

eduAristotlemetaphysicshtml

[] William Blake Songs of Experience Dover New York Fac-simile Reproduction with Plates in Full Color

[] Maxime Bocirccher On a nine-point conic Annals of Mathemat-

ics VI() January wwwjstororgstable1967142 ac-cessed November

[] Carl B Boyer A History of Mathematics John Wiley New York

[] Brianchon and Poncelet On the nine-point circle theorem In A

Source Book in Mathematics [] pages ndash Originally in Geor-gonnersquos Annales de Matheacutematiques vol (ndash) pp ndashTranslated from the French by Morris Miller Slotnick

[] Frances Marguerite Clarke and David Eugene Smith ldquoEssay pourles Coniquesrdquo of Blaise Pascal Isis ()ndash March Trans-lation by Clarke Introductory Note by Smith wwwjstororg

stable224736

[] R G Collingwood The Principles of Art Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood The Idea of Nature Oxford University PressLondon Oxford and New York paperback edition Firstpublished

[] R G Collingwood An Autobiography Clarendon Press Oxford First published With a new introduction by StephenToulmin Reprinted

[] R G Collingwood An Essay on Metaphysics Clarendon PressOxford revised edition With an Introduction and additionalmaterial edited by Rex Martin Published in paperback Firstedition

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

[] H S M Coxeter Introduction to Geometry John Wiley New Yorksecond edition First edition

[] H S M Coxeter The Real Projective Plane Springer New Yorkthird edition With an appendix for Mathematica by GeorgeBeck First edition

[] Reneacute Descartes The Geometry of Reneacute Descartes Dover Publica-tions New York Translated from the French and Latin byDavid Eugene Smith and Marcia L Latham with a facsimile of thefirst edition of

[] Reneacute Descartes The Philosophical Writings of Descartes volume ICambridge University Press translated by John CottinghamRobert Stoothoff and Dugald Murdoch

[] Reneacute Descartes La Geacuteomeacutetrie Jacques Gabay Sceaux France Reprint of Hermann edition of

[] Diogenes Laertius Lives of Eminent Philosophers Number inthe Loeb Classical Library Harvard University Press and WilliamHeinemann Cambridge MA and London Volume I of twoWith an English translation by R D Hicks First published Available from archiveorg and wwwperseustuftsedu

[] Euclid Euclidis Elementa volume I of Euclidis Opera Omnia Teub-ner Leipzig Edited with Latin interpretation by I L HeibergBooks IndashIV

[] Euclid The Thirteen Books of Euclidrsquos Elements Dover Publica-tions New York Translated from the text of Heiberg with in-troduction and commentary by Thomas L Heath In three volumesRepublication of the second edition of First edition

[] Euclid Στοιχεία Εὐκλείδου httpusersntuagrdimoureuclid Edited by Dimitrios E Mourmouras Accessed December

[] Euclid The Bones A handy where-to-find-it pocket reference com-

panion to Euclidrsquos Elements Green Lion Press Santa Fe NM Conceived designed and edited by Dana Densmore

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

[] Euclid Euclidrsquos Elements Green Lion Press Santa Fe NM All thirteen books complete in one volume The Thomas L Heathtranslation edited by Dana Densmore

[] Karl Wilhelm Feuerbach Eigenschaften einiger merkwuumlrdigen

Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten

Linien und Figuren Wiessner Nuumlrnberg Goumlttingen Stateand University Library gdzsubuni-goettingende accessedNovember

[] Karl Wilhelm Feuerbach On the theorem which bears his name InA Source Book in Mathematics [] pages ndash Originally in [](d ed ) Translated from the German by Roger A Johnson

[] David Fowler The Mathematics of Platorsquos Academy A new recon-

struction Clarendon Press Oxford second edition

[] Henri Frankfort Mrs H A Frankfort John A Wilson and Thork-ild Jacobsen Before Philosophy The Intellectual Adventure of An-

cient Man An Oriental Institute Essay Penguin Baltimore First published as The Intellectual Adventure of Ancient Man

Reprinted

[] Martin Gardner Fractal Music Hypercards and More Mathemati-

cal recreations from Scientific American magazine W H FreemanNew York Columns from the and issues

[] James Gow A Short History of Greek Mathematics UniversityPress Cambridge archiveorg accessed October

[] Timothy Gowers June Barrow-Green and Imre Leader editors The

Princeton Companion to Mathematics Princeton University PressPrinceton NJ

[] Robin Hartshorne Geometry Euclid and beyond UndergraduateTexts in Mathematics Springer-Verlag New York

[] Thomas Heath Aristarchus of Samos The Ancient CopernicusClarendon Press Oxford A history of Greek astronomy toAristarchus together with Aristarchusrsquos treatise on the sizes anddistances of the sun and moon a new Greek text with translationand commentary

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

[] Thomas Heath A History of Greek Mathematics Vol I From

Thales to Euclid Dover Publications New York Correctedreprint of the original

[] Thomas Heath A History of Greek Mathematics Vol II From

Aristarchus to Diophantus Dover Publications New York Corrected reprint of the original

[] Herodotus The Persian Wars Books IndashII volume of the Loeb

Classical Library Harvard University Press Cambridge MA andLondon Translation by A D Godley first published revised

[] Herodotus The Landmark Herodotus The Histories Anchor BooksNew York A new translation by Andrea L Purvis withMaps Annotations Appendices and Encyclopedic Index Editedby Robert B Strassler With an Introduction by Rosalind Thomas

[] David Hilbert The Foundations of Geometry Authorized transla-tion by E J Townsend Reprint edition The Open Court Publish-ing Co La Salle Ill Based on lectures ndash Translationcopyrighted Project Gutenberg edition released December (wwwgutenbergnet)

[] Immanuel Kant Critique of Pure Reason The Cambridge Editionof the Works of Kant Cambridge University Press Cambridge pa-perback edition Translated and edited by Paul Guyer andAllen W Wood First published

[] G S Kirk J E Raven and M Schofield The Presocratic Philoso-

phers Cambridge University Press second edition Reprinted First edition

[] Morris Kline Mathematical Thought from Ancient to Modern

Times Oxford University Press New York

[] Henry George Liddell and Robert Scott A Greek-English LexiconClarendon Press Oxford ldquoRevised and augmented throughoutby Sir Henry Stuart Jones with the assistance of Roderick McKenzieand with the cooperation of many scholars With a revised supple-mentrdquo First edition ninth edition

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

[] Nicholas Lobachevski The theory of parallels In Roberto Bonolaeditor Non-Euclidean Geometry A Critical and Historical Study

of its Development Dover New York Translated by GeorgeBruce Halstead Open Court edition

[] Reviel Netz The Shaping of Deduction in Greek Mathematics vol-ume of Ideas in Context Cambridge University Press Cambridge A study in cognitive history

[] Emin Oumlzdemir Accedilıklamalı Atasoumlzleri Soumlzluumlğuuml Remzi İstanbul Annotated dictionary of [Turkish] sayings

[] Pappus of Alexandria Pappus Alexandrini Collectionis Quae Su-

persunt volume II Weidmann Berlin E libris manu scriptisedidit Latina interpretatione et commentariis instruxit FridericusHultsch

[] Pappus of Alexandria Book of the Collection Part Introduc-

tion Text and Translation Springer Science+Business Media NewYork Edited With Translation and Commentary by Alexan-der Jones

[] Dimitris Patsopoulos and Tasos Patronis The theorem of ThalesA study of the naming of theorems in school geometry textbooksThe International Journal for the History of Mathematics Educa-

tion () wwwcomapcomhistoryjournalindexhtml ac-cessed September

[] Dan Pedoe Circles MAA Spectrum Mathematical Association ofAmerica Washington DC A mathematical view Revisedreprint of the edition with a biographical appendix on KarlFeuerbach by Laura Guggenbuhl Originally published

[] David Pierce Induction and recursion The De Morgan Jour-

nal ()ndash httpeducationlmsacuk201204

david-pierce-induction-and-recursion

[] David Pierce St Johnrsquos College The De Morgan Journal ()ndash educationlmsacukwp-contentuploads201202

st-johns-collegepdf accessed October

[] David Pierce Abscissas and ordinates J Humanist Math()ndash scholarshipclaremontedujhmvol5

iss114

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

[] David Pierce On the foundations of arithmetic in Euclid http

matmsgsuedutr~dpierceEuclid January

[] David Pierce On commensurability and symmetry J Human-

ist Math ()ndash scholarshipclaremontedujhm

vol7iss26

[] Plutarch Septem sapientium convivium Teubner Leipzig Perseus Digital Library dataperseusorgtextsurnctsgreekLittlg0007tlg079 accessed November

[] Plutarch Septem sapientium convivium penelopeuchicago

eduThayerERomanTextsPlutarchMoraliaDinner_of_the_

Sevenhtml May Edited by Bill Thayer Text of thetranslation by F C Babbitt on pp ndash of Vol II of the LoebClassical Library edition of the Moralia

[] Proclus A Commentary on the First Book of Euclidrsquos ElementsPrinceton Paperbacks Princeton University Press Princeton NJ Translated from the Greek and with an introduction and notesby Glenn R Morrow Reprint of the edition With a forewordby Ian Mueller

[] Lucio Russo The Forgotten Revolution How Science Was Born

in bc and Why It Had to Be Reborn Springer-Verlag Berlin Translated from the Italian original by Silvio Levy

[] George Salmon A Treatise on Conic Sections Longman BrownGreen and Longmans London Third edition revised and en-larged archiveorgdetailsatreatiseonconi00unkngoog ac-cessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London sixth edition archiveorgdetails

atreatiseonconi07salmgoog accessed November

[] George Salmon A Treatise on Conic Sections Longmans Greenand Co London tenth edition archiveorgdetails

atreatiseonconi09salmgoog accessed November

[] Charles Smith and Sophie Bryant Euclidrsquos Elements of Geometry

books IndashIV VI and XI Macmillan London

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography

[] David Eugene Smith A Source Book in Mathematics vols DoverPublications New York Unabridged republication of the firstedition published by McGraw-Hill

[] Rene Taton Lrsquo laquo Essay pour les Coniques raquo de PascalRevue drsquohistoire de science et de leurs applications ()ndash wwwperseefrwebrevueshomeprescriptarticlerhs_0048-7996_1955_num_8_1_3488

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol I From Thales to Euclid Number in theLoeb Classical Library Harvard University Press Cambridge Mass With an English translation by the editor

[] Ivor Thomas editor Selections Illustrating the History of Greek

Mathematics Vol II From Aristarchus to Pappus Number inthe Loeb Classical Library Harvard University Press CambridgeMass With an English translation by the editor

[] Arthur W Weeks and Jackson B Adkins A Course in Geometry

Plane and Solid Ginn and Company Lexington MA

[] Wikipedia contributors Pappusrsquos hexagon theorem In Wikipedia

The Free Encyclopedia enwikipediaorgwikiPappus27s_

hexagon_theorem Accessed December

[] Wikipedia contributors Thales In Wikipedia The Free Encyclope-

dia enwikipediaorgwikiThales Accessed December

Bibliography