A HISTORY OF 7T (PI)

Petr BeckmannElectrical Engineering Department,

University of Colorado

ST. MARTIN'S PRESS

New York

TO [;MUDLA,who never doubted the success of this book

Copyright © 1971 by THE GOLEM PRESSAll rights reserved. For information, write:St. Martin's Press, Inc., 175 Fifth Ave., New York, N.Y. 10010.Manufactured in the United States of AmericaLibrary of Congress Catalog Card Number: 74-32539

Preface

The history of 17 is a quaint little mirror of the history of man. It is thestory of men like Archimedes of Syracuse, whose method of calculat-ing 17 defied substantial improvement for some 1900 years; and it is alsothe story of a Cleveland businessman, who published a book in 1931announcing the grand discovery that 17 was exactly equal to 256/81,a value that the Egyptians had used some 4,000 years ago. It is the storyof human achievement at the University of Alexandria in the 3rdcentury B.C.; and it is also the story of human folly which mademediaeval bishops and crusaders set the torch to scientific librariesbecause they condemned their contents as works of the devil.

Being neither an historian nor a mathematician, I felt eminentlyqualified to write that story.

That remark is meant to be sarcastic, but there is a kernel of truth init. Not being an historian, I am not obliged to wear the mask ofdispassionate aloofness. History relates of certain men and institutionsthat I admire, and others that I detest; and in neither case have I hesi-tated to give vent to my opinions. However, I believe that facts andopinion are clearly separated in the following, so that the reader shouldrun no risk of being overly influenced by my tastes and prejudices.

Not being a mathematician, I am not obliged to complicate myexplanations by excessive mathematical rigor. It is my hope that thislittle book might stimulate non-mathematical readers to becomeinterested in mathematics, just as it is my hope that students of physicsand engineering might become interested in the history of the tools they

4 PREFACE

are using in their work. There are, however, two sure and all too welltried methods of how to make mathematics repugnant: One is tobrutalize the reader by assertions without proof; the other is to -hit himover the head with epsilonics and proofs of existence and unicity.I have tried to steer a middle course between the two.

A history of 17 containing only the bare facts and dates when who didwhat to 17 tends to be rather dull, and I thought it more interesting tomix in some of the background of the times in which 17 made progress.Sometimes I have strayed rather far afield, as in the case of the RomanEmpire and the Middle Ages; but I thought it just as important toexplore the times when 17 did not make any progress, and why it did notmake any.

The mathematical level of the book is flexible. The reader who findsthe mathematics too difficult in some places is urged to do what themathematician will do when he finds it too trivial: Skip it.

This book, small as it is, would not have been possible without thewholehearted cooperation of the staff of Golem Press, and I take thisopportunity to express my gratitude to every one of them. I am alsoindebted to the Archives Division of the Indiana State Library formaking available photostats of Bill 246, Indiana House of Representa-tives, 1897, and to the Cambridge University Press, Dover Publicationsand Litton Industries for granting permission to reproduce copyrightedmaterials without charge. Their courtesy is acknowledged in the notesaccompanying the individual figures.

I much enjoyed writing this book, and it is my sincere hope that thereader will enjoy reading it, too.

Boulder, ColoradoAugust 1970

Petr Beckmann

PREFACE

PREFACE TO THE SECOND EDITION

5

After all but calling Aristotle a dunce, spitting on the Roman Empire, and flipping mynose at some other highly esteemed institutions, I had braced myself for the reviews thatwould call this book the sick product ofan insolent ignoramus. My surprise was thereforeall the more pleasant when the reviews were very favorable, and the first edition went outof print in less than a year.

I am most grateful to the many readers who have written in to point out misprints anderrors, particularly to those who took me to task (quite rightly) for ignoring the recenthistory of evaluating 17 by digital computers. I have attempted to remedy thisshortcoming by adding a chapter on 17 in the computer age.

Mr. D.S. Candelaber of Golem Press had the bright idea of imprinting the end sheetsof the book with the first 10,000 decimal places of 17, and the American MathematicalSociety kindly gave permission to reproduce the first two pages of the computer print-outas published by Shanks and Wrench in 1962. A reprint ofthis work was very kindly madeavailable by one of the authors, Dr. John W. Wrench, Jr. To all of these, I would like toexpress my sincere thanks. I am also most grateful to all readers who have given me thebenefit of their comments. I am particularly indebted to Mr. Craige Schensted of AnnArbor, Michigan, and M. Jean Meeus of Erps-Kwerps, Belgium, for their detailed lists ofmisprints and errors in the first edition.

Boulder, ColoradoMay 1971

PREFACE TO THE THIRD EDITION

P.B.

Some more errors have been corrected and the type has been re-set for the thirdedition.

A Japanese translation of this book was published in 1973.Meanwhile, a disturbing trend away from science and toward the irrational has set in.

The aerospace industry has been all but dismantled. CoIIege enroIIment in the hardsciences and engineering has significantly dropped. The disoriented and the guIlible flockin droves to the various Maharajas of Mumbo Jumbo. Ecology, once a respected scientificdiscipline, has become the buzzword of frustrated housewives on messianic ego-trips.Technology has wounded affluent intellectuals with the ultimate insult: They cannotunderstand it any more.

Ignorance, anti-scientific and anti-technology sentiment have always provided thebreeding ground for tyrannies in the past. The power of the ancient emperors, themediaeval Church, the Sun Kings, the State with a capital S, was always rooted in theignorance of the oppressed. Anti-scientific and anti-technology sentiment is providing abreeding ground for encroaching on the individual's freedoms now. A new tyranny is onthe horizon. It masquerades under the meaningless name of "Society."

Those who have not learned the lessons of history are destined to relive it.Must the rest of us relive it, too?

Boulder, ColoradoChristmas 1974

P.B.

Contents

1. DAWN 92. THE BELT 203. THE EARLY GREEKS 364. EUCLID 455. THE ROMAN PEST 556. ARCHIMEDES OF SyRACUSE 627. DUSK 738. NIGHT 789. AWAKENING 87

10. THE DIGIT HUNTERS 9911. THE LAST ARCHIMEDEANS 11012. PRELUDE TO BREAKTHROUGH 12113. NEWTON 13414. EULER. " 14715. THE MONTE CARLO METHOD 15816. THE TRANSCENDENCE OF 17 ••••••••••••••••••••••• 16617. THE MODERN CIRCLE SQUARERS 17318. THE COMPUTER AGE 183

Notes 190Bibliography 193Chronological Table 196Index 198

DAWNHistory records the names of royal

bastards. but cannot tell us the origin ofwheat.

JEAN HENRI FABRE(1823-1915)

13million years or so have passed since the tool-wielding animal~ called man made its appearance on this planet. During thistime it learned to recognize shapes and directions; to grasp theconcepts of magnitude and number; to measure; and to realize thatthere exist relationships between certain magnitudes.

The details of this process are unknown. The first dim flash in thedarkness goes back to the stone age - the bone of a wolf with incisionsto form a tally stick (see .figure on next page). The flashes becomebrighter and more numerous as time goes on, but not until about 2,000B.C. do the hard facts start to emerge by direct documentation ratherthan by circumstantial evidence. And one of these facts is this: By2,000 B.C., men had grasped the significance of the constant that istoday denoted by TT, and that they had found a rough approximationof its value.

How had they arrived at this point? To answer this question, wemust return into the stone age and beyond, and into the realm ofspeculation.

Long before the invention of the wheel, man must have learned toidentify the peculiarly regular shape of the circle. He saw it in the pupilsof his fellow men and fellow animals; he saw it bounding the disks ofthe Moon and Sun; he saw it, or something near it, in some flowers; andperhaps he was pleased by its infinite symmetry as he drew its shape inthe sand with a stick.

Then, one might speculate, men began to grasp the concept ofmagnitude - there were large circles and small circles, tall trees andsmall trees, heavy stones, heavier stones, very heavy stones. Thetransition from these qualitative statements to quantitative measure-

10 CHAPTER ONE

-...

A stone age tally stick. The tibia(shin) of a wolf with two longincisions in the center. and twoseries of 25 and 30 marks.Found in Vestonice. Moravia(Czechoslovakia) in 1937.2

ment was the dawn of mathematics. Itmust have been a long and arduous road,but it is a safe guess that it was first takenfor quantities that assume only integralvalues - people, animals, trees, stones,sticks. For counting is a quantitativemeasurement: The measurement of theamount of a multitude of items.

Man first learned to count to two, anda long time elapsed before he learned tocount to higher numbers. There is a fairamount of evidence for this, l perhapsnone of it more fascinating than thatpreserved in man's languages: In Czech,until the Middle Ages, there used to betwo kinds of plural - one for two items,another for many (more than two) items,and apparently in Finnish this is so tothis day. There is evidently no connectionbetween the (Germanic) words two andhalf; there is none in the Romance lan-guages (French: deux and moitie> nor inthe Slavic languages (Russian: dva andpo!), and in Hungarian, which is not anIndo-European language, the words arekettlJ and f~l. Yet in all European lan-guages, the words for 3 and 113,4 and 114, etc., are related. This suggeststhat men grasped the concept of a ratio,and the idea of a relation between anumber and its reciprocal, only after theyhad learned to count beyond two.

The next step was to discover relationsbetween various magnitudes. Again, itseems certain that such relations werefirst expressed qualitatively. It must havebeen noticed that bigger stones are heavier, or to put it into morecomplicated words, that there is a relation between the volume and theweight of a stone. It must have been observed that an older tree is taller,that a faster runner covers a longer distance, that more prey gives morefood, that larger fields yield bigger crops. Among all these kinds of

DAWN 11

relationships, there was one which could hardly have escaped notice,and which, moreover, had no exceptions:

The wider a circle is "across." the longer it is "around."And again, this line of qualitative reasoning must have been followed

by quantitative considerations. If the volume of a stone is doubled, theweight is doubled; if you run twice as fast, you cover double thedistance; if you treble the fields, you treble the crop; if you double thediameter of a circle, you double its circumference. Of course, the ruledoes not always work: A tree twice as old is not twice as tall. The reasonis that "the more ... the more" does not always imply proportionality;or in more snobbish words, not every monotonic function is linear.

Neolithic man was hardly concerned with monotonic functions; but itis certain that men learned to recognize, consciously or unconsciously,by experience, instinct, reasoning, or all of these, the concept ofproportionality; that is, they learned to recognize pairs of magnitudesuch that if the one was doubled, trebled, quadrupled, halved or leftalone, then the other would also double, treble, quadruple, halve orshow no change.

And then carne the great discovery. By recognizing certain specificproperties, and by defining them, little is accomplished. (That is whythe old type of descriptive biology was so barren.) But a great scientificdiscovery has been made when the observations are generalized in sucha way that a generally valid rule can be stated. The greater its range ofvalidity, the greater its significance. To say that one field will feed halfthe tribe, two fields will field the whole tribe, three fields will feed oneand a half tribes, all this applies only to certain fields and tribes. To saythat one bee has six legs, three bees have eighteen legs, etc., is astatement that applies, at best, to the class of insects.But somewherealong the line some inquisitive and smart individuals must have seensomething in common in the behavior of the magnitudes in these andsimilar statements:

No matter how the two proportional quantities are varied. their ratioremains constant.

For the fields, this constant is 1 : 112 = 2 : 1 = 3 : Ph = 2. For thebees, this constant is 1 : 6 = 3 : 18 = 1/6. And thus, man had dis-covered a general, not a specific, truth.

This constant ratio was not obtained by numerical division (andcertainly not by the use of Arabic numerals, as above); more likely, theratio was expressed geometrically, for geometry was the first mathe-matical discipline to make substantial progress. But the actual tech-

12 CHAPTER ONE

nique of arriving at the constancy of the ratio of two proportionalquantities makes little difference to the argument.

There were of course many intermediate steps, such as the discoveryof sums, differences, products and ratios; and the step of abstraction,exemplified by the transition from the statement "two birds and twobirds make four birds" to the statement "two and two is four." Butthe decisive and great step on the road to 17 was the discovery thatproportional quantities have a constant ratio.

From here it was but a dwarfs step to the constant 17: If the"around" (circumference) and the "across" (diameter) of a circlewere recognized as proportional quantities, as they easily must havebeen, then it immediately follows that the ratio

circumference : diameter = constant for all circles.This constant circle ratio was not denoted by the symbol 17 until

the 18th century (A.D.), nor, for that matter, did the equal sign ( = )come into general use before the 16th century A.D. (The twin linesas an equal sign were used by the English physician and mathemati-cian Robert Recorde in 1557 with the charming explanation that"noe .2. thynges, can be moare equalle.") However, we shall usemodern notation from the outset, so that the definition of the num-ber 17 reads

C17 =

D

where C is the circumference, and D the diameter of any circle.And with this, our speculative road has reached, about 2,000 B.C.,

the dawn of the documented history of mathematics. From thedocuments of that time it is evident that by then the Babyloniansand the Egyptians (at least) were aware of the existence and signi-ficance of the constant TT as given by (1).

BUT the Babylonians and the Egyptians knew more about 17than its mere existence. They had also found its approximate value.By about 2,000 B.C., the Babylonians had arrived at the value

and the Egyptians at the value

(2)

17 (3)

DAWN

c

EaA~...L-----'--~'-'------'--I---"4----J

BE

How to measure 7T in the sandsoftheNile

13

How did these ancient people arrive at these values? Nobodyknows for certain, but this time the guessing is fairly easy.

Obviously, the easiest way is to take a circle, to measure itscircumference and diameter, and to find 7T as the ratio of the two.Let us try to do just that, imagining that we are in Egypt in 3,000B.C. There is no National Bureau of Standards; no calibrated meas-uring tapes. Weare not allowed to use the decimal system ornumerical division of any kind. No compasses, no pencil, no paper;all we have is stakes, ropes and sand.

So we find a fairly flat patch of wet sand along the Nile, drive in astake, attach a piece of rope to it by loop and knot, tie the other endto another stake with a sharp point, and keeping the rope taut, wedraw a circle in the sand. We pull out the central stake, leaving ahole 0 (see figure above). Now we take a longer piece of rope, chooseany point A on the circle and stretch the rope from A across the holeo until it intersects the circle at B. We mark the length AB on therope (with charcoal); this is the diameter of the circle and our unit oflength. Now we take the rope and lay it into the circular groove inthe sand, starting at A. The charcoal mark is at C; we have laid offthe diameter along the circumference once. Then we lay it off asecond time from C to D, and a third time from D to A, so that thediameter goes into the circumference three (plus a little bit) times.

If, to start with, we neglect the little bit, we have, to the nearestinteger,

17=3 (3)

To improve our approximation, we next measure the little left-overbit EA as a fraction of our unit distance AB. We measure the curved

14 CHAPTER ONE

23 ~n i'~~ ~:0M~ ~J~

~~~ TD~'" ~~~~ I t,~~ 'M~~ 'M~~~ ~~;t~~;~ '~N ~~: ~~~ Q~~~ mm 'M~'i?

28. Kat (TrOCT/af TTfV {)at..aaaav OfKa fV 1TTfXH aTTO TOU XHt..OU"; aUT7Jt;((,It; TOV t/iHt..OVt; aUTTfS, aTpOyyUt..OV KUft..W TO aUTO. TTfVTf fV TTTfXH TO

VI/10F, aVTTfS' Kat aVvTfYllfVTf TpHS "at TpWKOvra fV TTtXH.

21 Hizo asimismo un mar de fundi-cion. de diez codos del uno al otrolado. redondo. y de cinco codos dealto. y cefiialo en derredor un cor-don de treinta codos.

2:l. II fit uussi une mer de fonte, dedix coudeeH d'un bord jusqu'a I'autre,qui ctuit toutc ronde: eUe avalt cinql'OUd«(PH de haut, et elle etait envi-ronndc tout h I'cntour d'un cordon detn'ntl' coudees.

23. IId15lal tet mafe slite, deslti laket ad jednaha kraje k druhemu,o!uollhlJ l'ukal. a pet laket byla uysakast jeha, a okolek jeha tFicetI "J.>e I 1'17kal.

23. Unb er mad)te ein 9Jtecr, gegoffen,non einem 9\anb ~um anbern 5d)tt (f({enroeit, runbum~er, unb fiinf ~Uen ~od),nnb eine 6d)nur breiBig ~Uen lang warbas 9JtaB ringsum.

23. And he made a molten sea, ten cubits from the one brimto the other; it was round all about, and his height was fivecubits: and a line of thirty cubits did compass it round about.

DAWN 15

length EA and mark it on a piece of rope. Then we straighten therope and lay it off along AB as many times as it will go. It will gointo our unit distance AB between 7 and 8 times. (Actually, if weswindle a little and check by 20th century arithmetic, we find that 7is much nearer the right value than 8, i.e., that E 7 in the figure onp. 13 is nearer to B than E s , for 1/7 = 0.142857 , 118 = 0.125,and the former value is nearer 17 - 3 = 0.141592 However, thatwould be difficult to ascertain by our measurement using thick,elastic ropes with coarse charcoal marks for the roughly circularcurve in the sand whose surface was judged "flat" by arbitraryopinion.)

We have thus measured the length of the arc EA to be between1/7 and 118 of the unit distance AB; and our second approximationis therefore

(4)

for this, to the nearest simple fractions, is how often the unit ropelength AB goes into the circumference ABeD.

And indeed, the values

7T

are the values most often met in antiquity.For example, in the Old Testament (I Kings vii.23, and 2 Chro-

nicles iv.2), we find the following verse:

"Also, he made a molten sea of ten cubits frombrim to brim. round in compass, and five cubits theheight thereof; and a line of thirty cubits did compassit round about."

The molten sea, we are told, is round; it measures 30 cubits roundabout (in circumference) and 10 cubits from brim to brim (in dia-meter); thus the biblical value of 17 is 30/10 = 3.

The Book of Kings was edited by the ancient Jews as a religiouswork about 5SO B.C., but its sources date back several centuries. Atthat time, 7T was already known to a considerably better accuracy,but evidently not to the editors of the Bible. The Jewish Talmud,which is essentially a commentary on the Old Testament, was pub-lished about SOO A.D. Even at this late date it also states "thatwhich in circumference is three hands broad is one hand broad."

16 CHAPTER ONE

The molten sea as reconstructed by Gressmanfrom the description in 2 Kings vii.'

In early antiquity, in Egypt and other places, the priests wereoften closely connected with mathematics (as custodians of the calen-dar, and for other reasons to be discussed later). But as the processof specialization in society continued, science and religion driftedapart. By the time the Old Testament was edited, the two werealready separated. The inaccuracy of the biblical value of 17 is, ofcourse, no more than an amusing curiosity. Nevertheless, with thehindsight of what happened afterwards, it is interesting to note thislittle pebble on the road to confrontation between science and re-ligion, which on several occasions broke out into open conflict, andabout which we shall have more to say later.

Returning to the determination of 17 by direct measurement usingprimitive equipment, it can probably safely be said that it led tovalues no better than (4).

From now on, man had to rely on his wits rather than on ropesand stakes in the sand. And it was by his wits, rather than byexperimental measurement, that he found the circle's area.

THE ancient peoples had rules for calculating the area of acircle. Again, we do not know how they derived them (except for onemethod used in Egypt, to be described in the next chapter), and once

DAWN

Calculation of the area of a circle byintegral calculus. The area of an element-ary ring is ciA =211pdp; hence the area ofthe circle is

rA = 211 .rpdp 11r 2 •

(J

17

more we have to play the game "How do you do it with theirknowledge" to make a guess. The area of a circle, we know, is

A = 11r (5)

where r is the radius of the circle. Most of us first learned thisformula in school with the justification that teacher said so, take itor leave it, but you better take it and learn it by heart; the formulais, in fact, an example of the brutality with which mathematics isoften taught to the innocent. Those who later take a course in theintegral calculus learn that the derivation of (5) is quite easy (seefigure above). But how did people calculate the area of a circlealmost five millenia before the integral calculus was invented?

They probably did it by a method of rearrangement. They cal-culated the area of a rectangle as length times width. To calculatethe area of a parallelogram, they could construct a rectangle of equalarea by rearrangement as in the figure below, and thus they foundthat the area of a parallelogram is given by base times height. Theage of rigor that came with the later Greeks was still far away; they

The paraIIelogram and the rectangle have equal areas,as seen by cutting off the shaded triangle and reinsert-

ing it as indicated.

18 CHAPTER ONE

2rrr

2Trr

(hi(a)

2 Tr r

2 Trr-_..._------------------(

DAWN 19

The rearrangement methodused in a 17th century Japa-

nese document. 4

much more closely; and the area of the circle is still exactlly one halfof the quasi-parallelogram (c).

On continuing this process by cutting up the original circle into alarger and larger number of segments, the side formed by the littlearcs of the segments will become indistinguishable from a straightline, and the quasi-parallelogram will turn into a true parallelogram(a rectangle) with sides 2 TTr and r. Hence the area of the circle ishalf of this rectangle, or 1Tr2•

The same construction can be seen in the Japanese documentabove (1698). Leonardo da Vinci also used this method in the 16thcentury. He did not have much of a mathematical education, and inany case, he could use little else, for Europe in his day, debilitatedby more than a millenium of Roman Empire and Roman Church,was on a mathematical level close to that achieved in ancient Meso-potamia. It seems probable, then, that this was the way in whichancient peoples found the area of the circle.

And that should be our last speculation. From now on, we can relyon recorded history.

THE BELT

Accurate reckoning - the entrance intothe knowledge ofall existing things and allobscure secrets.

AHMES THE SCRIBE17th century B.C.

II)AN is not the only animal that uses tools on his environ-ment; so do chimpanzees and other apes (also, some birds).As long as man was a hunter, the differences between thenaked ape and the hairy apes was not very radical. But roughlyabout 10,000 B.C., the naked ape learned to raise crops and to tameother animals, and thereby he achieved something truly revolution-ary: Human communities could, on an average, produce so muchmore food above the subsistence minimum that they could free apart of their number for activities not directly related to the provi-sion of food and shelter.

This Great Agricultural Revolution first took place where thegeographical conditions were favorable: Not in the north, where thewinters were long and severe, and the conditions for farming gener-ally adverse; nor in the tropics, where food was plentiful, clothingunnecessary, shelter easily available, and therefore no drastic needfor improvement; but in the intermediate belt, where conditions weresufficiently adverse to create pressures for change, yet not so adverseas to foil the attempts of farming and livestock raising.

This intermediate belt stretched from the Mediterranean to thePacific. The Great Revolution first took place in the big river valleysof Mesopotamia; later the Belt stretched from Egypt through Persiaand India to China. States developed. Specialists came into being.Soldiers. Priests. Administrators. Traders. Craftsmen. Educators.

And Mathematicians.

THE BELT 21

The hunters had neither time nor need for ratios, proportionalitiesor conic sections. The new society needed surveyors and builders,navigators and timekeepers (astronomers), accountants and stock-keepers, planners and tax collectors, and, yes, mumbo-jumbo men toimpress and bamboozle the uneducated and oppressed. This was thefertile ground in which mathematics flourished; and it is thereforenot surprising that the cradle of mathematics stood in this Belt.

SINCE Mesopotamia was the first region of the Belt where theagricultural revolution occurred and a new society took hold, onewould expect Babylonian mathematics to be the first and mostadvanced. This was indeed the case; the older literature on thehistory of mathematics often saw the Egyptians as the founders ofmathematics, but this was due to the fact that more and earlierEgyptian documents than any others used to be available. The re-search of the last few decades has changed this, and as a smallsidelight, we find a better approximation for 17 in Mesopotamia thanin Egypt.

One of the activities for which the new society freed some of itsmembers was, unfortunately, organized warfare, and the variouspeoples inhabiting the region at different times, Sumerians, Baby-lonians, Assyrians, Chaldeans and others, warred against each otheras well as against outsiders such as Hittites, Scythians, Medes andPersians. The city of Babylon was not at all times the center of thisculture, but the mathematics coming from this region is simplylumped together as "Babylonian."

In 1936, a tablet was excavated some 200 miles from Babylon.Here one should interject that the Sumerians were first to make oneof man's greatest inventions, namely, writing; through written com-munication, knowledge could be passed from one person to others,and from one generation to the next and future ones. They im-press~d their cuneiform (wedge-shaped) script on soft clay tabletswith a stylus, and the tablets were then hardened in the sun. Thementioned tablet, whose translation was partially published only in1950,5 is devoted to various geometrical figures, and states that theratio of the perimeter of a regular hexagon to the circumference ofthe circumscribed circle equals a number which in modern notationis given by 57/60 + 36/(60)2 (the Babylonians used the sexagesimalsystem, i.e., their base was 60 rather than 10).

22 CHAPTER TWO

rr

r cThe Babylonian value of TT.

The Babylonians knew, of course, that the perimeter of a hexagonis exactly equal to six times the radius of the circumscribed circle, infact that was evidently the reason why they chose to divide the circleinto 360 degrees (and we are still burdened with that figure to thisday). The tablet, therefore, gives the ratio 6r/C, where r is theradius, and C the circumference of the circumscribed circle. Usingthe definition TT = C/2r, we thus have

which yields

3TT

57

60+

i.e., the value the Babylonians must have used for TT to arrive at theratio given in the tablet. This is the lower limit of our little thoughtexperiment on p. 15, and a slight underestimation of the true valueof TT.

MORE is known about Egyptian mathematics than about themathematics of other ancient peoples of the pre-Hellenic period. Notbecause they had more of it, nor because more of their documentshave been found, but because the key to the Egyptian hieroglyphswas discovered much earlier than for the other cultures. In 1799, theNapoleonic expedition to Egypt found a trilingual tablet at Rosettanear Alexandria, the so-called Rosetta stone. Its message was re-corded in Greek, Demotic and Hieroglyphic. Since Greek was known,the code was cracked, and decipherment of the Egyptian hieroglyphs

THE BELT 23

proceeded rapidly during the last century. The Babylonian tabletsare more durable, and tens of thousands have been unearthed; theuniversity libraries of Columbia and Yale, for example, have largecollections. However, an analogous trilingual stone in Persian, Me-dean and Assyrian was not deciphered until about 100 years ago(with Persian known, and its writing system related to Babylonian),and as far as the history of mathematics is concerned, substantialprogress in deciphering tablets in cuneiform script was not madeuntil the 1930's. Even now, a large quantity of the available tabletsawait investigation.

The oldest Egyptian document relating to mathematics, and forthat matter, the oldest mathematical document from anywhere, is apapyrus roll called the Rhind Papyrus or the Ahmes Papyrus. It wasfound at Thebes in a room of a ruined building and bought by aScottish antiquary, Henry Rhind, in a Nile resort town in 1858. Fouryears later, it was purchased from his estate by the British Museum,where it is now, except for a few fragments which unexpectedlyturned up in 1922 in a collection of medical papers in New York,and which are now in the Brooklyn Museum.

Histories like these make one wonder how many such pricelessdocuments have been used up by the Arabs as toilet papyri. The sadstory of how some of these papyri and tablets have survived milleniaonly to be rendered worthless by the excavators of our time is told byNeugebauer. 6

The Ahmes Papyrus contains 84 problems and their solutions (butoften no hint on how the solution was found). The papyrus is a copyof an earlier work and begins thus: 7

Accurate reckoning. The entrance into the knowledge of all exist-ing things and all obscure secrets. This book was copied in the year33, in the 4th month of the inundation season. under the King ofUpper and Lower Egypt "A-user-Re." endowed with life. in likenessto writings made ofold in the time of the King of Upper and LowerEgypt Ne-mat'et-Re. It is the scribe Ahmes who copies this writing.

From this, Egyptologists tell us, we know that Ahmes copied the"book" in about 1650 B.C. The reference to Ne-mat'et-Re dates theoriginal to between 2,000 and 1800 B.C., and it is possible that someof this knowledge may have been handed down from Imhotep, theman who supervised the building of the pyramids around 3,000 B.C.

A typical problem (no. 24) runs like this:A heap and its 1/7 part become 19. What is the heap?The worked solution accompanying the problem is this:

24 CHAPTER TWO

Assume 7.(Then) 1 (heap is) 7(and) 117 (of the heap is) 1(making a) Total 8(But this is not the right answer, and therefore)As many times as 8 must be multiplied to give 19, so many times

7 must be multiplied to give the required number.The text then finds (by rather complicated arithmetic) the solution

7 x 191s = 16510.Today we would solve the problem by the equation

x+xl7 = 19,

where x is the "heap." The Egyptian method, called regula falsi, i.e.,assuming a (probably wrong) solution and then correcting it byproportionality, is no longer used in high school algebra. However,one version of regula falsi, called scaling, is still very advantageousfor some electrical circuits (see figure below).

But the problem of interest here is no. SO. Here Ahmes assumesthat the area of a circular field with a diameter of 9 units is thesame as the area of a square with a side of 8 units. Using theformula for the area of a circle A = 17r, this yields

17 (9/2)2 = 82,

and hence the Egyptian value of 17 was

17 = 4 x (8/9) 2 = 3.16049... ,

20th century version of Abmes' method. The calculation of the currentsdue to a known voltage V in the network above becomes verycumbersome when the circuit is worked from left to right. So oneassumes a (probably wrong) current of 1 ampere in the last branch andworks the circuit from right to left (which is much easier); this ends upwith the wrong voltage, which is then corrected and all currents arescaled in proportion. Just as was done by the Egyptians in 2,000 B.C.

The circuit above is also a mathematical curiosity for another reason.Ifall its elements are 1 ohm resistors, and the current in the last branchis 1 ampere, then the voltages across the resistors (from right to left) areFibonacci numbers (1, I, 2, 3, 5, 8, 13, 21, 34, 55, ...• each newmember being the sum of the last two).

THE BELT 25

Egyptian method of calculating 17.

a value only very slightly worse than the Mesopotamian value 3 118,and in contrast to the latter, an overestimation. The Egyptian valueis much closer to 3 116 than to 3 1/7, suggesting that it was neitherobtained nor checked by experimental measurement (which, as onp. 13, would have given a value between 3 1/7 and 3 118).

How did the Egyptians arrive at this weird number? Ahmes pro-vides a hint in Problem 48. Here the relation between a circle andthe circumscribed square is compared. Ahmes forms an (irregular)octagon by trisecting the sides of a square with length 9 units (seefigure above) and cutting off the comer triangles as shown. The areaof the octagon ABCDEFGH does not differ much from the area ofthe circle inscribed in the square, and equals the area of the fiveshaded squares of 9 square units each, plus the four triangles of 4112square units each. This is a total of 63 square units, and this is closeto 64 or 82 • Thus the area of a circle with diameter 9 approximatelyequals 64 square units, i.e., the area of a square with side 8, which,as before, will lead to the value 17 = 4 x (8/9) 2.

Ahmes has, of course, swindled twice: once in setting the area ofthe octagon equal to the area of the circle, and again in setting63 '" 64. It is, however, noteworthy that these two approximationspartially compensate for each other. Indeed, for a square of side a,the area of the octagon is 7(aI3)2 , and this is to equal p2, where p isthe side of the square with equal area. If we now swindle only onceby setting the area of the octagon equal to that of the circle, then

1,417a2 = p2 = 7(aI3)2,

and from the first and last expressions we find 17 = 2819 = 3 119.This is a much worse approximation that Ahmes', confirming thetime-honored rule that the last error in a calculation should cancelout all the preceding ones.

26 CHAPTER TWO

bE agricultural revolution in the Indian river valleys pro-bably took place at roughly the same time as along the Nile valleyand the valley of the Euphrates and Tigris (Mesopotamia). Theapparent delay of Indian mathematics behind that of Babylon, Egyptand Greece may well be simply due to our ignorance of early Indianhistory. There is much indirect evidence of Hindu mathematics. Likeeverybody else in the Belt, they knew Pythagoras' Theorem longbefore Pythagoras was born, and there is evidence that Hindu astro-nomy was at a highly advanced level. Direct records, however, havebeen lost, and the first available documents are the Siddhantas orsystems (of astronomy), published about 400 A.D., though the know-ledge contained in them is of course much older.

One of the Siddhantas, published in 380 A.D., uses the value

17 = 3 177/1250 = 3.1416

which differs little from the sexagesimal value

17 = 3 + 8/60 + 30/(60)2

used by the Greeks much earlier.Early Hindu knowledge was summarized by Aryabhata in the

Aryabhatiya, written in 499 A.D. This gives the solutions to manyproblems, but usually without a hint of how they were found. Onestatement is the following: 8

Add 4 to 100. multiply by 8, and add 62,000. The result isapproximately the circumference of a circle of which the diameteris 20.000.

This makes

17 = 62,832 : 20,000 = 3.1416

as in the Siddhanta. The same value is also given by Bashkara (born1114 A.DJ, who calls the above value "exact," in contrast to the"inexact" value 3 1/7.

It is highly likely9 that the Hindus arrived at the value above bythe Archimedean method of polygons, to which we shall come inChapter 6. If the length of the side of a regular polygon with n sidesinscribed in a circle is s(n), then the corresponding length for 2nsides is

THE BELT 27

s (2n) = )2 - J 4 - s2(n)Starting (naturally) with a hexagon, progressive doubling leads to

polygons of 12, 24, 48, 96, 192 and 384 sides. On setting thediameter of the circle equal to 100, the perimeter of the polygon with384 sides is found to be the square root of 98694, whence

TT '" V98694/100 = 3.14156... ,

which is the value given by Arayabatha.The Hindu mathematician Brahmagupta (born 598 A.D.) uses the

valueTT '" ylO = 3.162277...

y965, y981, y986 and y987,

the Hindus may have (incorrectly) assumed that on increasing thenumber of sides, the perimeter would ever more closely approach thevalue yl000, so that

which is probably also based on Archimedean polygons. It has beensuggested 10 that since the perimeters of polygons with 12, 24, 48 and96 sides, inscribed in a circle with diameter 10, are given by thesequence

TT = J1000 110 = Yl0.

WHAT has been said about the ancient history of mathe-matics in India, and our ignorance of it, applies equally well to thePacific end of the Belt, China. However, study of this subject hasrecently been facilitated by the impressive work of Needham. 11

As in the West (see p. 14), the value TT '" 3 was used for severalcenturies; in 130 A.D., Hou Han Shu used TT = 3.1622, which isclose to TT = Y10. (The Chinese were singular among the ancientpeoples in that they used the decimal system from the very be-ginning.) A document in 718 A.D. takes TT = 92/29 = 3.1724 ... LiuHui (see figure on next page), in 264 A.D., used a variation of theArchimedean inscribed polygon; using a polygon of 192 sides, hefound

3.141024 < TT < 3.142704

and with a polygon of 3,072 sides, he found TT = 3.14159.

jE.• 1

III m ~

I:\

'J

.~ ~J

'..

~

~~~

~~

~

~

18th century explanation of Liu Hui'smethod (264 A.D.) of calculating the

approximate value of 17. 12

Determination of the diameter and cir-cumference of a walled city from a distant

observation point (1247 A.D.). 13

THE BELT 29

In the 5th century, Tsu Chung-Chih and his son Tsu Keng-Chihfound

3.1415926 < 17 < 3.1415927,

an accuracy that was not attained in Europe until the 16th century.Not too much should be made of this, however. The number of

decimal places to which 17 could be calculated was, from Archimedesonward, purely a matter of computational ability and perseverance.Some years ago, it was only a matter of computer programmingknow-how; and today it is, in principle, no more than a matter ofdollars that one is willing to spend for computer time. The importantpoint is that the Chinese, like Archimedes, had found a method that,in principle, enabled them to cakulate TT to any desired degree ofaccuracy.

Yet the high degree of accuracy that the Chinese attained issignificant in demonstrating that they were far better equipped fornumerical calculations than their western contemporaries. The rea-son was not that they used the decimal system; the decimal systemby itself proves nothing except that nature was a poor mathematicianin giving us ten fingers (instead of a number that has more integralfactors, like twelve). As human language shows, everybody used baseten for numeration (or its multiple, 20, as in French and Danish).

But the Chinese discovered the equivalent of the digit zero. Likethe Babylonians, they wrote numbers by digits multiplying powers ofthe base (10 in China, 60 in Mesopotamia), just like we do. Butwhere the corresponding power of 10 was missing (102 has no tens),they left a space. The Hindus later used a circle for the digit zero (0),and this reached Europe via the Arabs and Moors only in the lateMiddle Ages (Italy) and the early Renaissance (Britain). An edict of1259 A.D. forbade the bankers of Florence to use the infidel sym-bols, and the University of Padua in 1348 ordered that price lists ofbooks should not be prepared in "ciphers," but in "plain" letters(i.e., Roman numerals).14 But until the infidel digit 0 was imported,few men in Europe had mastered the art of multiplication anddivision, let alone the extraction of square roots which was needed tocalculate 17 in the Archimedean way.

~E Belt, as we have argued, was the region conducive to theGreat Agricultural Revolution that turned human communities frompacks of hunters to societies with surplus productivity, freeing some

30 CHAPTER TWO

of its members for activities other than provision of food. If this Beltstretched from the valley of the Nile to the Pacific, why should it notalso cross America?

Indeed, it did. The agricultural revolution had taken place, attimes comparable to the origins of the Afro-Asian Belt, in parts ofCentral and South America, where the impressive civilizations of theAztecs, the Maya, the Chibcha, the Inca and others had grown up.A Maya ceremonial center in Guatemala, by carbon 14 tests, datesback to 1182 ( ± 240) B.C,15 and agriculture had of course begunmuch earlier; in fact, it was the American Indian who first domesti-cated two of our most important crops - corn (maize) and thepotato. 16

The most advanced of these cultures was that of the Maya, whoestablished themselves in the Yucatan peninsula of Mexico, parts ofGuatemala, and western Honduras. Judging by the Maya calendar,Maya astronomy must have been as good as that of early Egypt, forby the first century A.D. (from which time we have some dateddocuments), they had developed a remarkably accurate calendar,based on an ingenious intermeshing of the periods of the Sun, theMoon and the Great Star noh ek (Venus). The relationship betweenthe lunar calendar and the day count was highly accurate: The erroramounted to less than 5 minutes per year. The Julian calendar,which had been introduced in Rome in the preceding century, andwhich some countries of the Old World retained up to the 20thcentury, was in error by more than 11 minutes per year. From thisalone it does not follow that the Maya calendar was more accuratethan the Julian calendar (as some historians have concluded), but thepoint has some bearing, albeit very circumstantial, on the value thatthe Maya might have used for IT, and we shall digress to examine thehistory of our calendar.

A calendar is a time keeping device. If it is to be of practical usebeyond a prediction of when to observe a religious holiday, it mustbe matched to the seasons of the year (i.e., to the earth's orbit roundthe sun) very accurately: If we lose an hour or two per calendar yearwith respect to the time it takes the earth to return to the same pointon its orbit, this does not seem a significant error; yet the differencewill accumulate as the years go by, and eventually one will find thatat 12 noon on a summer day (by this calendar) it is not only dark,but it is snowing as well.

Astronomy for calendar making was therefore one of the earliestactivities involving mathematics in all ancient societies. In Egypt

THE BELT 31

and Babylon, as well as in Maya society, the priests had a monopolyof learning. In all three societies the priests were the astronomers,calendar makers and time keepers.

The Babylonians first took a year equal to 360 days (presumablybecause of their sexagesimal system and their 3600 circle), but laterthey corrected this by 5 additional days. This value was also adoptedby the Egyptians, and it was to be the value used by the Maya. Tocorrect this value to a fraction of a day, it was necessary to observethe movement of the other stars on the celestial sphere. The Mayawatched the planet Venus, the Egyptians the fixed star Sirius. Owingto the precession and nutation of the earth's axis, Sirius is not reallyfixed on the celestial sphere (tied to the coordinates of the terrestrialobserver), but has, as viewed from the earth, a motion of its own.The Egyptians found that Sirius moved exactly one day ahead every4 years, and this enabled them to determine the length of one yearas 365114 days. One of the Greco-Egyptian rulers of Egypt in theHellenic age, Ptolemy III Euergetes, a mathematician whom we shallmeet again, issued the following decree in 238 B.C.:

Since the Star [SiriusIadvances one day every four years, and in orderthat the holidays celebrated in the summer shall not fall into winter, ashas been and will be the case if the year continues to have 360 and 5additional days, it is hereby decreed that henceforth every four yearsthere shall be celebrated the holidays ofthe Gods of Euergetes after the5 additional days and before the new year, so that everyone might knowthat the former shortcomings in reconing the seasons of the year havehenceforth been truly corrected by King Euergetes. 17

But by this time Egyptian priesthood had become more interestedin religious mumbo jumbo than in science, and they sabotaged theenforcement of Ptolemy's decree. Ironically, it fell to the Romanadventurer, warlord and vandal Gaius Julius Caesar to bring aboutthe adoption of the leap year. Alexandria, in the 3rd to 1st centuriesB.C., was the intellectual center of the ancient world, the like ofwhich had not been seen before and was not to be seen again untilthe rise of Cambridge and the Sorbonne. In 47 B.C., Caesar's hordesransacked the city and burned its libraries, and Caesar, duringwhatever time he had left between his romance with Cleopatra andcontemplating further conquests, managed to get acquainted withthe astounding achievements of Alexandrian astronomers. He tookone of them, Sosigenes, back to Rome and inaugurated the Juliancalendar as of January 1, 45 B.C. The new calendar introducedPtolemy's leap year, giving a year an average duration of 365.25

32 CHAPTER TWO

days. But the true duration of the earth's orbit about the sun isabout 365.2422 days. (Actually, the earth is subject to all kinds ofperturbations, and today we do not use its orbit as a standard anymore. "Atomic clocks" run much more accurately than the earth.When such an atomic clock appears to be x seconds late at the endof a tropical year, we say that the earth has completed its orbit xseconds early.)

The difference between the two values, a little over 11 minutes peryear, accumulated over the centuries and once more threatened toput the date out of date. In 1582, Pope Gregory XIII decreed thatthe extra day of a leap year was to be omitted in years that aredivisible by 100, unless also divisible by 400 (i.e., omitted in 1800,1900, but not in 2(00). This is the calendar we are using now. It wassoon adopted by the Catholic countries, but the others thought it"better to disagree with the Sun than tn agree with the Pope," andBritain, for example, did not adopt it until 1752. By that time theBritish had slipped behind 11 days, and when they were simplyomitted to catch up with the rest of Europe, many Britons wereoutraged, accusing the government of conspiring to shorten theirlives by 11 days and to rob them of the interest on their bankaccounts. Russia held on to the Julian calendar until the OctoberRevolution, which took place in November (1917).

Getting further and further away from the Maya, we may notethat our present calendar is far from satisfactory. Not because it isstill off by 2 seconds per year, for these accumulate to a full day onlyonce in 3,300 years; but because our present calendar is highlyirregular: A date falls on a different day of the week every year, andthe months (even the quarters) have different lengths, which com-plicates, among other things, accounting and other business admi-nistration. In 1923 the League of Nations established a Committeefor Calendar Reform, which, predictably, achieved nothing, and thesame result has hitherto been achieved by the United Nations. Of thehundreds of submitted proposals, UNESCO in 1954 recommendedthe so-called "World Calendar" for consideration by the UN GeneralAssembly. The proposal does away with both of the mentioneddisadvantages of our present calendar, yet does not change it soradically as to cause widespread confusion. Most governments agreedto the proposal in principle, but some (including the US) consideredit "premature" and the matter is still being "considered." The UN,a grotesque assembly of propaganda-bent hacks, has found itselfunable to condemn international terrorism by criminals, much less toreform the calendar.

THE BELT 33

Maya glyph denotingposition of month in half-

year period.

It is against this background of difficul-ties with calendar making that the achieve-ments of the Maya must be viewed. True,their calendar year, like that of the Baby-lonians and Egyptians, amounted to 365days, so that their religious festivals driftedwith respect to the natural seasons. Like theBabylonians and Egyptians, they must haveknown that they were gaining one day everyfour years, but like the Egyptians, they evi-dently preferred keeping their religious holi-days intact to using the calendar as a timekeeper for sowing, harvesting and other activities geared to theperiodicity of nature.

If we do not require a calendar to be geared to a tropical year(earth's orbit), but only that it be geared to some part of the celestialclock, then the Maya calendar was more accurate than the Juliancalendar, more accurate than the Egyptian (solar) calendar, andmore accurate than the Babylonian (solar-lunar) calendar; it inter-meshed the "gear wheels" of Sun, Moon and Venus, and was basedon a more accurate "gear ratio" than the other calendars, repeatingitself only once in 52 years.

It is unthinkable that a people as advanced in astronomy shouldnot have come across the problem of calculating the circle ratio. Iftheir astronomy was as advanced as that of the Egyptians, is it notreasonable to assume that the Maya value of 17 was as good as thatof the Egyptians?

It is not. It is reasonable to assume that it was far better. For theMaya were incomparably better equipped for numerical calculationsthan the Egyptians were; they had discovered the zero digit and thepositional notation that had escaped the genius of Archimedes, andthat held up European arithmetic for a thousand years after theMaya were familiar with it. They used the vigesimal system (base 20),the digits from 1 to 19 being formed by combinations of ones (dots)and fives (bars), as shown in the figure below.

••• ••• - -1 3 5 7

••• •••• Maya digits- - - ~- - -- - -15 18 19 0

34 CHAPTER TWO

•

••-

57 216,340

••--~-•

~

20

•---••322

•• • •--- 20'20·

Maya numerical notation(read from top to bottom)

Any positionally expressed number abcd.ef using base x denotesthe number

The Maya notation fits this pattern with x = 20 and the digitsshown on p. 33, with an exception for the second-order digit. In apure vigesimal system, this would multiply 2()2 = 400, but for rea-sons connected with their calendar, the Maya used this position tomultiply the number 18 x 20 or 2()2 - 40. Also, it is not knownwhether they progressed beyond a "vigesimal point" to negativepowers of 20, i.e. to vigesimal fractions. Examples of the Mayanotation are given in the figure above. However this may be, it isclear that with a positional notation closely resembling our own oftoday, the Maya could out-calculate the Egyptians, the Babylonians,the Greeks, and all Europeans up to the Renaissance.

The Chinese, who had also discovered the digit zero and the posi-tional notation utilizing it, had found the value of 11 to 8 significantfigures a thousand years before any European. The Maya valuemight have been close to that order.

But we can only guess. The Maya civilization went the way ofother civilizations; after it peaked, it decayed. About the 7th centuryA.D., the Maya started to desert their temple cities, and within acentut:y or two these splendid cities were abandoned, no one knows

THE BELT 35

why. Civil strife set in, and later they were conquered by the Aztecs.By the time the Spaniards arrived, they were far down from theirclassical age.

There were some records, of course. The Maya wrote books onlong strips of bark or parchment, folded like a screen. How many ofthese dealt with their mathematics, geometry and the circle ratio, weshall never know. In the 1560's, Diego de Landa, Bishop of Yucatan,burned the literature of the Maya on the grounds that "they con-tained nothing in which there were not to be seen superstition andlies of the devil." 18 What remained was burned by the natives whohad been converted to the Bishop's religion of love and tolerance.

Today the American Indians, in their quest for identity and self-respect, complain that even the name of their people was given them"by some honkey who landed here by mistake."

They might add that the Red Man had made the great discoveryof positional notation employing the digit zero a thousand yearsbefore the Palefaces; at a time when Spain was a colony of abenighted empire, and when the ancestors of the Anglo-Saxons wereilliterate hunters in the virgin forests of continental Europe, no oneknows exactly where.

The Early Greeks

o King, for traveling over the country,there are royal roads and roads for com-mon citizens; but in geometry there is oneroad for all.

MENAECHMUS (4th century B.C.),when his pupil Alexander the Great askedfor a shortcut to geometry.

U N following the Belt along its length, we have lost the threadof time; and we now return to the Eastern Mediterranean,where the history of mankind went through a few inspiringcenturies associated with the ancient Greeks. The Golden Age of thistime was the age of the University of Alexandria; but just as theAlexandrian sun continued to glimmer some centuries after theRomans had sacked its places of learning and burned its libraries, sothere was already some light before the city and its University werefounded.

This was a time when the Greeks first held their own against thethen all-mighty Persian Empire at Marathon (490 B.C.) and thendefeated the Persians at Plataea (479 B.C.). This was also the timewhen democratic government developed; a slaveowners' democracy,yes. but a democracy. The next 150 years saw the confrontation ofAthens and Sparta, the thinkers against the thugs. The thugs alwayswin. but the thinkers always outlast them.

In mathematics, which is but a mirror of the society in which itthrives or suffers. the pre-Athenian period is one of colorful men andimportant discoveries. Sparta, like most militaristic states before andafter it, produced nothing. Athens. and the allied Ionians, produceda number of works by philosophers and mathematicians; some good,some controversial, some grossly overrated.

THE EARLY GREEKS 37

As far as the history of 17 is concerned, there were four men of thisperiod who had some bearing on the problem: Anaxagoras, Anti-phon, Hippocrates and Hippias.

Anaxagoras of Qazomenae (500-428 B.C.) found in Athens a boldnew spirit of free enquiry and enthusiastically joined in with a trulyscientific spirit, not shying away from experiment nor from popular-izing science (both in diametrical opposition to the snobbism of thelater Greek philosophers). He taught that the annual swelling of theNile was due to the melting of mountain snows near the upper partof the river; that the Moon received its light solely from the Sun;that eclipses of the Moon or the Sun were caused by the inter-position of the Earth or the Moon; that the Sun was a red-hot stonebigger than the entire Peloponnese; and other (less successful) theo-ries. But in his theory of the Sun, which denied that the Sun was adeity, he had gone too far, and for some time he was imprisoned inAthens for impiety.

While in jail, he attempted to "square the circle," a problemintimately connected with 17, which requires the construction of asquare equal in area to a given circle, but which we shall examine inmore detail in the next chapter. It is apparently the first mention (byPlutarch) of this famous problem that fascinated men until 1882,when the proof that it could not be solved (by Greek geometry) wasfinally furnished; and even now it continues to fascinate:: amateurcircle squarers who cannot be dissuaded by mathematical proof.

Another Greek of this period (late 5th century B.C.) associatedwith squaring the circle is the Sophist philosopher Antiphon, whoenunciated the "principle of exhaustion," which was to have a pro-found influence on mathematicians in their quest for the value of TTuntil the invention of the calculus in the late 17th century. Theexhaustion principle states the following: If a square is inscribed in acircle, and further regular polygons are inscribed with double thenumber of sides at each step (octagon, 16-sided polygon, etc.), untilthe circle is exhausted, then eventually a polygon will be reachedwhose sides are so short that it will coincide with the circle. This ledAntiphon to believe that the circle could be squared by Greekgeometry; for any polygon can be squared, and since the "eventual"polygon is equivalent to a circle, Antiphon judged that a circle couldbe squared also.

Antiphon's principle is almost correct; it becomes completely cor-rect in Euclid's more cautious formulation, according to which thedifference between the remaining area and that of the circle can be

38 CHAPTER THREE

A

Hippocrates lune. The lower case lettersstand for areas.

made smaller than any preassigned value, no matter how small (butnot zero).

Another man associated with the area of circles during this periodwas Hippocrates of Chios, who should not be confused with hisbetter known contemporary, the physician Hippocrates of Cos. Hip-pocrates of Chios was a merchant who came to Athens in 430 B.C.According to some, he lost his money through fraud in Byzantium,according to others he was robbed by pirates; in any case, hethereafter turned to the study of geometry, which is in itself remark-able, for in later years the road between business and science wasmore often traveled in the opposite direction.

Hippocrates was evidently the first to find the exact area of afigure bounded by curves, in this case circular arcs, and thus becamethe first man to make a precise statement on curvilinear mensura-tion. The area that Hippocrates squared (or triangled, but to squarea triangle is trivial) was that of a lune (see figure above) bounded bythe semicircular arc ABC circumscribed about the right-angled,isosceles triangle ABC, and the circular arc ADC such that thesegment ADEC is similar to the segment AB. (That is, the radius ofADC is to A C as the radius EA is to AB.)

Hippocrates found that the area of this lune is exactly equal to thetriangle ABC.

Proof: Since the segments above the three sides are similar, theirareas are to each other as the squares of their bases; since byPythagoras' Thorem AB2 + BO = AO, the areas of the corre-sponding segments are related by

t = u + w.

The area of the lune is

1= u + v + w,

THE EARLY GREEKS 39

Examples of figures that are exactly squarable by Euclidean geometry.The shaded areas are reducable to Hippocrates lunes, and hence are

squarable.

or on substituting (1),I = v + t,

i.e., it is equal to the area of the triangle.This is probably not the proof given by Hippocrates, whose book

on geometry has been lost. The proof of this theorem given byEudoxos and taken over by Euclid more than a century later is areductio ad absurdum (reduction to the absurd), a proof by contra-diction or indirect proof. It is possible that Hippocrates was the firstto use this method of proof, whose essence is this: If we want toprove that a statement is true, we first assume that it is not true anduse this assumption to achieve an absurd. result (or a result thatcontradicts the assumption); since the result is absurd, the premisemust have been false; if a statement is either true or false (tertiumnon datur, there is no third possibility), the statement must be true.

For example, the fact that the number of primes (numbers divis-ible only by themselves or one) is infinite is proved as follows.Assume that the number of primes is finite. Then there must existthe largest prime, call it p. Now the number

pI = p(p - 1) (p - 2) (p - 3) ... 2 . 1is divisible by all integers up to and including p. Hence the numberpI + 1 is divisible by none of them and is therefore a prime. But theprime pI + 1 is obviously greater than p, which contradicts theassumption that p is the largest prime. The assumption is thereforefalse, and the number of primes is infinite.

Hippocrates' discovery can easily be generalized to many othercurvilinear figures, two of which are shown above. The one on theright, in particular, must have raised high hopes of squaring thecircle (or the semicircle, which is just as good). The snag, as we knowtoday, is that some, but not all, lunes can be squared. In the top

40 CHAPTER THREE

right figure, for example, the sum of the lune and the semicircle canbe squared, but they cannot be squared individually - at least not bythe Greek rules of the game (which will be discussed in the nextchapter).

Very many other figures composed entirely or partly of Hippocrateslunes can be constructed. Leonardo da Vinci (1452-1519) was particu-larly fascinated by Hippocrates lunes, and he constructed 176 suchfigures on a single page of his manuscripts, and more elsewhere. 19

It is ironic that the work of Antiphon and Hippocrates should havebeen brought to our attention by one whose teachings held up theprogress of science for close to 2,000 years, the Greek philosopherAristotle (384-322 B.C.). He judges both Antiphon's principle of ex-haustion and Hippocrates' quadratures to be false, considering therefutation of Antiphon's principle beneath the notice of geometers,and never, of course, achieving a disproof of Hippocrates' quadratureof the lune, either. Aristotle, we are invaribly told, was "antiquity'smost brilliant intellect," and the explanation of this weird assertion, Ibelieve, is best summarized in Anatole France's words: The booksthat everybody admires are the books that nobody reads. But ontaking the trouble to delve in Aristotle's writings, a somewhat differ-ent picture emerges (see also Chapter 6). His ignorance of mathe-matics and physics, compared to the Greeks of his time, far surpassesthe ignorance exhibited by this tireless and tiresome writer in themany. subjects that he felt himself called upon to discuss.

THE fourth man of interest in this early Greek period is Hippiasof Elis, who came to Athens in the second half of the 5th century B.C.,and who was the first man on record to define a curve beyond thestraight line and circle. It is perhaps ironic that the next curve on thelist should be a transcendental one, skipping infinitely many algebraiccurves, but the Greeks did not yet know about degrees of a curve, andso they ate fruit from all orchards. The curve that Hippias discoveredwas called a trisectrix, because it could be used to trisect an angle (thesecond problem that fascinated antiquity, the third being the doublingof the volume of a cube), or a quadratrix, because it could be used tosquare the circle, and with the mere use of compasses and straight-edge, too. There has to be a snag, of course, and the snag was that theconstruction violated another rule of the Greek game, but this point isdeferred to the next chapter.

THE EARLY GREEKS

B P C,------------=;=--,--------------,

41

A oHippias' quadratrix

D

Hippias' quadratrix was defined as follows. Let the straight segmentAB move uniformly from the indicated position until it coincides withCD, and let the segment AO rotate uniformly clockwise about thepoint 0 from the indicated position through OP until it coincides withOD, in the same time as AB moves to CD. The curve traced by theintersection (L) of the two segments during this motion is Hippias'quadratrix.

Using only compasses and straightedge, and discarding the con-cepts of motion and time, we can also construct this curve by dividingthe angle AOP into 2n parts, and the segment AO into the samenumber of parts, where n is arbitrarily large. Any intersection of twocorresponding segments is then a point on the quadratrix.

We do not know whether Hippias realized that by means of hiscurve the circle could be squared; perhaps he realized it, but couldnot prove it. The proof was later given by Pappus (late 3rd centuryA.D.), who evidently had it from Dinostratus (born about 350 B.C.,brother of Menaechmus, who is quoted at the beginning of thischapter). His proof by elementary geometry is a reductio ad ab-surdum, but the modern road is shorter:

Let OL = p and OA = r; from the definition of the quadratrix,and expressing the angle of rotation in radian measure, we have

AU17/2 - e = const =

2r

17

42

and since AL' = r

CHAPTER THREE

p sin e, we obtain

(1)2r = 1Tp(sine)/e.

ForO -> O,we have (sine)/e -> 1, p -> OQ; also 2r=AD, so that(1) yields

AD : OQ = 1T,

and we have a geometrical construction for the circle ratio 1T.To rectify the circle, we need a segment of length

u = 27Tr = 2(AD:OQ)xOA = (AD x AD): OQ

or

u : AD = AD: OQ,

(2)

so that u can be found· by an easy construction of proportionalquantites (similar triangles).

A rectangle with sides ul2 and r then has area

Le., the same area as a circle with radius r, and since a rectangle iseasily squared by an elementary construction, the circle is squared bythe use of compasses and straightedge alone. Nevertheless, as we shallsee, this construction failed to qualify under the implicit rules ofGreek geometry.

H IPPIAS and Dinostratus only scratched the surface of thiscurve, for analytical geometry was still some 2,000 years away. But wemight as well finish the job for them.

Let us regard (1) as the equation of the quadratrix in polar co-ordinates through

P V(x2 + y2),

¢ arctan(y/x);

THE EARLY GREEKS

+ 17th century-00 « 00

43

i i

-x

Hippias' quadratrix derived by analytical geometry.Hippias knew only the part in the interval - a < x < a.

i.e.• the one shown in the preceding figure.

After some manipulation, we have

y = x cot( Tr x / 2a ) , (3)

and the figure above shows how the quadratrix has bloomed. Hippiasand Dinostratus only saw the tip of the iceberg for - I' < X < r.

The circle can also be squared by the use of the Archimedeanspiral, as we shall see.

The essence of Hippias' trick was the fact that the limit of (sin e )Iefor e ... 0 is unity. But there are other curves exploiting this limit, andthese can also be used as quadratrices. Three more appeared in the17th century after Descartes' discovery of analytical geometry:

Tschirnhausen's quadratrix,

Y = sin( Tr x / 2I' ) ,Ozam's quadratrix

Y = 21' sin2 (x/2r) ,

and the cochleoid, which in polar coordinates is given byP = (I' sin ¢)/¢.

44 CHAPTER THREE

But the problem has lost its fascination, and these curves have, formost people, faded into oblivion. However, when viewed as functionsrather than curves, they are still making a living in electromagnetics,spectral analysis and information theory, occasionally moonlighting instatistics.

EUCLIDKing Edward's new policy of peace wasvery successful and culminated in theGreat War to End War. [It was followedby] the Peace to End Peace.

w.e. Sellar and R.I. Yeatman,1066 and All That 20

fjj OMETHING in the vein of the quotation above might also besaid of Alexander the Great. He certainly had no policy ofpeace, but he, too, demonstrated the French saying that "themore this changes, the more it is the same thing." At age 32 he hadconquered the world, at least the world known to antiquity, everythingthere was to conquer between Greece and India, and some more intothe bargain. At age 33 he was dead, and so was his great empire: itbroke up into a heap of little empires, each of which wanted to be theGreat Empire and therefore fought all the other little empires.

During his expedition to Egypt in 332-331 B.C., Alexander hadfounded the city of Alexandria. After his death in 323 B.C., hisgenerals fought each other over who was to get his hands on what, butby 306 B.C. control over Egypt had firmly been established by one ofthem, Ptolemy I; he was succeeded by his son Ptolemy II, who in 276B.C. married his own sister Arsinoe. To marry one's own sister was aPharaonic practice (though the Ptolemies were Greeks); but Ptolemydid something much more significant. He founded the Museum andLibrary of Alexandria. 21

For the Library, Ptolemy acquired the most valuable manuscripts,had translations made of them, had purchasing agents scour theMediterranean for valued books, and even compelled travelers arriv-

46 CHAPTER FOUR

ing in Egypt to give up any books in their possession; they were copiedby scribes in the Library, the original was retained, and the copy givento its owner. His son Ptolemy III(the one who decreed leap years, secp. 31) was even more avid: He borrowed the original copies of theworks of famous Greek playwrights from the Athenians, had themcopied, sent back the copies and cheerfully forfeited the deposit hehad paid as bond for the return of the originals. 22 Before the arrivalof Caesar's thugs, the Library had close to three quarters of a millionbooks (that is, rolls; a standard roll was 15 to 20 feet in length andcontained the equivalent of ten to twenty thousand words of modernEnglish text22 ).

Even more impressive was the Museum. This was a school orinstitute - in effect, a university. Ptolemy engaged the most cele-brated scholars of his time to teach at this university, and soon itbecame the scientific capital of the world. Few were the learned menof later antiquity who had not studied at Alexandria; they were taughtby the finest scientists that the contemporary world could muster.

Mathematics flourished at Alexandria. Erastosthenes (273-192B.C.), chief librarian, calculated the circumference of the earth towithin SOJa of the correct value by observing the difference in zenithdistance of the sun at two places separated by a known distance(Alexandria and Syene, on approximately the same meridian); inposession of ever more accurate trigonometrical tables (obtained byworking down from 90° by use of the half-angle formulas and thentilling in the gaps by the addition theorems), he calculated thedistance to the moon and to the sun. The method was correct, andalthough his imperfect measuring instruments yielded a large error forthe moon, the ditance to the sun, as near as we can ascertain thelength of his unit, the stadium, agrees with what we know today,including measurements by radar. And this was done at a time whenthe philosopher Epicurus in Athens taught that the sun was two feetin diameter!

The academic community at Alexandria was cosmopolitan, mainlyGreek, Egyptian and Jewish. So was the city surrounding it. It wasreferred to as "Alexandria near Egypt," for it was not considered apart of Egypt proper. Indeed, it was not part of anything, evolving aculture of its own. Its citizens had a reputation of being lively,quick-witted, irreverent, and worthless as soldiers. The first threePtolemies were unusually enlightened, but later decadence set in, andthe line ended with the Quisling Queen Cleopatra. At times theAlexandrians would riot against a king they disliked and some of the

~ledarilfun.. ltbtl'd(JlJaJtolum I£udidiG perfpil~t:tn alUm 4.leomctrieillfipitquifodiCiIlimc:

tfl1(tuGeft(muG ~lG nodl.«Iliool eft:l"situdo fine lantudmecut'} quidc q.1tremtratc6 ft ouopueta. «Ilmcarcaa

f~,~~ ,c.~ '9IlOpu~o adaltu blC!liHima~(Iho 1qrremwrcs fUBG -etrUq;eo, redpicntl.«I0uRficiescq,lOsltudini1lartrudtnc trii b;:cui9tmnl quidc fUt linee.«I0uRfici«lplana cab 'COO bnca ad QIltaqtCtio i qrremttltCG fUaG redptc6

It'~~~I«I21nsutu6planUG' oua~ linCQru all~,~~~~.•~~~, temu6l'tQctU6:qua¥qpaho (fUR fURlftclC applkanoq;nootreeta.«I~ado au!anSUlum ?tmct Ouebnce rccre rectibnc'9ansulue noiaf. «I JOn recta linea fUR rectliftetmtbooq; inSUlt«robtq;.fucrit e~la;J:'eo, ~q; rect9crtt«Iltncaq;ltneefu~ae ri (ot tURftat~pcndtcuLart6 -eoclif.«I21nsuluG~b qui recto maio: cobtutbe Oldt.«I21n5ul'}~0 mmo: rccro acot9appdlaf.«IXmntn9cq6 ~iuf(utu(q; futiG c.«I;i5ura,q imino -errmnfu?rind.«IJ,(trcul9cfisura plana -ena qdcm lit

From a Latin edition of Euclid's Elements (1482).62

DcpMdpij~Rfc Ilotw:oz plna IX oltfini,tiombus carolldem.

1.fnta

~

I fUI!nac. plallll. I

1\0'" !f... "R ~~ a Ii.~ 5' lil• JiO,"Cfrcula. \.. i· in

t"1c::::Rt3

"'"'"I

48 CHAPTER FOUR

later kings would order a massacre. That, they must have laughed,would cure them. They are still laughing.

AMONG the scholars whom Ptolemy brought to Alexandria wasEuclid, a man whose place and date of birth are unknown, so today heis simply called "Euclid of Alexandria." Euclid was, among otherthings, a publisher's dream. His Elements (of geometry) are the all-time bestseller of all textbooks ever written. More than a thousandeditions have been published only since the invention of the letter-press in the 15th century, and the textbook, now in (roughly) its2,2SOth year, is still going strong (especially considering that practi-cally all school geometries are merely rehashes of Euclid's text).

The major part of the Elements was probably known before Euclid.But the importance of Euclid's work is not in what the theorems, assuch, say. The great significance of the Elements lies in their method.The Elements are the first grandiose building of mathematical archi-tecture. There are five foundation stones, or postulates, which (Euclidbelieved) are so simple and obvious that everyone can accept them.Onto these foundation stones Euclid lays brick after brick with ironlogic, making sure that each new brick rests firmly supported by onepreviously laid, with not the tiniest gap a microbe can walk through,until the whole cathedral stands as firmly anchored as its foundations.

Euclid is not the father of geometry; he is the father of mathema-tical rigor.

The Elements consist of twelve books, and the elementary geometrytaught in the world's high schools corresponds, in one form oranother, to the first four. Alas, the reason why it should be taught isvery often grossly misunderstood. Except for a few elementary theo-rems, Euclidean geometry is of little use for modern science andengineering: One can usually go much further and faster by trigo-nometry and analytical geometry. The real signifiance of Euclideangeometry lies in the superb training it gives for logical thinking.A proof must not contain anything that is ultimately based onwhat we want to prove, or the proof is circular and invalid. From"Every angle in a semicircle is a right angle" it does not follow that"The apex of a right angle subtended by the diameter lies on thecircumference;" it so happens that the second statement is true, but ithas to be proved. Euclidean geometry teaches the difference betweenifand ifand only if, and the difference between one and one and only

EUCLID 49

one. If somebody teaches you how to bisect an angle by the use ofcompasses without insisting on the proof that the construction iscorrect, he has thrown out the golden kernel and handed you the deadgarbage. Who needs to bisect an angle, anyway? The few who do finda protractor faster and more accurate than using compasses threetimes over. But this is not the point; what matters is that theconstructed line is the bisector and the only bisec~or.

And this brings us to the much misunderstood problem of squaringthe circle, i.e., constructing a square whose area equals that of agiven circle. There never was an Olympic Committee that laid downthe rules of the game and the conditions under which the circle was tobe squared. Yet from the way Euclid's cathedral is built, we knowexactly what the Greeks had in mind. The problem was this:

(l) Square the circle,(2) using straightedge and compasses only,(3) in a finite number of steps.It is evident why Hippias' and Dinostratus' method of squaring the

circle was not accepted as "clean" by the Greeks, even thoughDinostratus proved that it did square the circle. Not because itviolated the second condition; it did not. Nor (as some write) becausethe quadratrix was a curve other than a straight line or a circle:Euclid himself wrote a book on conic sections (lost in the Romancatastrophe), and these were generally not circles, either. But referringto the figure on p. 41, it is easily seen that the point Q can only befound by placing a French curve along the points L, which violatesrule 2, or constructing infinitely many points L by straightedge andcompasses. Even assuming that the Greeks would have intuitivelyaccepted the equivalent of the modern concept of continuity (and theyaccepted nothing but a handful of postulates), they meticulouslysteered clear of anything involving the infinite, for they were stillpuzzled by Zeno's paradoxes (about SOO B.C.), the best known ofwhich is the one of Achilles and the tortoise: Achilles races thetortoise, giving it a headstart of 10 feet. While Achilles covers these 10feet, the tortoise covers 1 foot; Achilles covers the remaining foot, butthe elusive tortoise has covered another 1/10 of a foot, and so on;Achilles can never catch up with the tortoise. The stumbling blockhere was that the Greeks could not conceive of an infinite sum addingup to a finite number. In our own age, school children absorb the factthat this is so at the tender age when they first learn about decimalfractions:

10/9 = 1.1111 ... = 1 + 1110 + 11100 + 111000 + 1110000 + ...

50 CHAPTER FOUR

As for the condition that only compasses and straightedge may beused in the construction, it has puzzled many, including some whoought to know better. It has been attributed to Greek esthetics and tothe influence of Plato and Aristotle (from whom the Alexandrianscientists were mercifully removed by an ocean far larger than theMediterranean); and other reasons have been suggested to explain the"Greek obsession with straightedge and compasses."

Nonsense. The only things the Greek mathematicians were obsessedwith were truth and logic. Straightedge and compasses came into theproblem only incidentally and as a secondary consequence, andesthetics had nothing to do with it at all. Just as Euclid built hiscathedral on five foundation stones whose simplicity made themobvious, so one can go in the opposite direction and prove a statementby demonstrating that it was entirely supported by the stones ofEuclid's building; from hereon downward it had already been provedthat the building stones lie as firmly anchored as the foundations.Proving the correctness of an assertion, therefore, amounted to reduc-ing it to the "obvious" validity of the foundation stones.

Euclid's five foundation stones, or "obvious" axioms, were thefollowing:

I. A straight line may be drawn from any point to any other point.II. A finite straight line may be extended continuously in a straight

line.III. A circle may be described with any center and any radius.IV. All right angles are equal to one another.The fifth postulate is unpleasantly complicated, but the general

idea is also conveyed by the following formulation (not used by Euclid,whose axioms do not contain the concept parallel):

V. Given a line and a point not on that line, there is not more thanone line which can be drawn through the point parallel to the originalline.

To prove a statement (such as that a construction attains a givenobject> meant the following to the Greeks: Reduce the statement toone or more of the axioms above, i.e., make the statement as obviousas these axioms are.

It is easily seen that Euclid's axioms are visualizations of the mostelementary geometrical constructions, and that they are accomplishedby straightedge and compasses. (This is where the straightedge andand compasses come in as an entirely subordinate consideration.) It isevident that if a construction uses more than straightedge and com-passes (e.g., a French curve), it can never be reduced to the five

EUCLID 51

Euclidean axioms, that is, it could never be proved in the eyes of theancient Greeks. 23 That, and nothing else, was the reason why theGreeks insisted on straightedge and compasses.

Let us illustrate this by a specific example, the squaring of arectangle. Weare required to construct a square whose area is equalto the area of a given rectangle ABCD.

A(b)F(a)

D

A

11III7How to square a rectangle. Building stone no. I1I.3S.

Construction (see Figure a above): Extend AB, and make BE = BC.Bisect AE to obtain 0, and with 0 as center, describe a circle withradius OA. Extend CB to intersect the circle at F. Then BF is the sideof the required square.

Proof Let G be the intersection of BC extended and the circle APE.Then by Euclid III.35 (Book III, Theorem 35, to be quoted below),

AB x BE=FB x BG.But by Euclid m.3,

FB=BGand by construction,

BE=BCHence,

AB X BC=FB2 ,as was to be demonstrated.

The two theorems referred to above areII!.35. If two chords intersect within a circle, the rectangle con-

tained by the segments of one equals the rectangle contained by thesegments of the other.

II!.3. If the diameter ofa circle bisects a chord, it is perpendicularto it; or, if perpendicular to it, it bisects it.

This would usually be all that is required for the proof of theconstruction, for both theorems are proved by Euclid. However, let us

52 CHAPTER FOUR

trace the path back to the axioms. Our proof rests, among others, onbuilding stone no. BI.35, which says (figure b on p. 51) that

AB x BE = CB x BF.This rests, among others, on the theorem that similar triangles

(CAB and BFE) have proportional sides. To show that these trianglesare similar, we have to show, among other things, that the angles CABand BFE are equal. This theorem, in turn, rests on the theorem thatthe angle at the circumference equals half the angle at the center,subtended by the same chord. This, in turn, rests on the theorem thatthe angles at the base of an isosceles triangle are equal. This againrests on the theorem that if one triangle has two sides and theincluded angle equal to two sides and the included angle of another,then the two triangles are congruent. And the congruency theoremsfinally rest almost immediately on the Axioms.

We have traced but a single path from stone no. III.35 down to thefoundation stones of Euclid's cathedral. This path has many otherbranches (branching off where it says "among others" above), and ifwe had the patience, these could be traced down to the foundationstones, too. So could the paths from stone no. III.3 and from theoperations used in the construction (whose feasibility need not betaken for granted).

Thus, the proof that the construction for squaring a rectangle asshown in the figure on p. 51 is correct, is essentially a reduction toEuclid's axioms.

No such reduction is possible with Hippias' construction for squar-ing the circle. For the reasons given above, it cannot find a restingplace in Euclid's cathedral; the paths of proof (such as the derivationgiven on pp. 41-42) will not lead to Euclid's foundation stones, andHippias' stone will tumble to the ground.

This does not mean that Hippias' construction is incorrect. Hippias'stone can find rest in a cathedral built on different foundations. WhatLindemann proved in 1882 was not that the squaring of the circle (orits rectification, or a geometrical construction of the number 17) wasimpossible; what he proved (in effect) was that it could not be reducedto the five Euclidean axioms.

LET us now take a brief look at the later fate of these fivefoundation stones, the five Euclidean axioms or postulates. Therewere a few flaws in them which were later mended, but we have no

EUCLID 53

c

BWhat does

"straight" mean?

time for hairsplitting here. We will take a look ata more significant aspect.

The attitude of the ancient Greeks to Euclid-ean geometry was essentially this: "The truth ofthese five axioms is obvious; therefore everythingthat follows from them is valid also."

The attitude of modern mathematics is some-what different: "If we assume that these axiomsare valid, then everything that follows from themis valid also."

At first sight the difference between the twoseems to be a chicken-hearted technicality. Butin reality it goes much deeper. In the 19th cen-tury it was discovered that if the fifth postulate (p. SO) was pulled outfrom under Euclid's cathedral, not all of the building would collapse;a part of the structure (called absolute geometry) would remainsupported by the other four axioms. It was also found that if the fifthaxiom was replaced by its exact opposite, namely, that it is possible todraw more than one straight line through a point parallel to a givenstraight line, then on this strange fifth foundation stone (together withthe preceding four) one could build all kinds of weird and wonderfulcathedrals. Riemann, Lobachevsky, Bolyai and others built just suchcrazy cathedrals; they are known as non-Euclidean geometry.

The non-Euclidean axiom may sound ridiculous. But an axiom isunprovable; if we could prove it, it would not be an axiom, for it couldbe based on a more primitive (unprovable) axiom. We just assume itsvalidity or we don't; all we ask of an axiom is that it does not lead tocontradictory consequences. And non-Euclidean geometry is just asfree of contradictions as Euclidean is. One is no more "true" than theother. The fact that we cannot draw those parallel lines in the usualway proves nothing.

Nevertheless, some readers may feel that all this is pure mathema-tical abstraction with no relation to reality. Not quite. Reality is whatis confirmed by our