Cajori - The History of Zeno's Arguments on Motion - Phases in the Development of the Theory of Limits

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The History of Zeno's Arguments on Motion: Phases in the Development of the Theory ofLimitsAuthor(s): Florian CajoriSource: The American Mathematical Monthly, Vol. 22, No. 3 (Mar., 1915), pp. 77-82Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2971890.

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THEAMERICAN

MATHEMATICAL MONTHLYVOLUMEXXII MARCH,1915 NUMBER3

THE HISTORY OF ZENO'S ARGUMENTS ON MOTION.PHASES IN THE DEVELOPMENT F THE THEORYOF LIMITS.

IV.By FLORIAN CAJORI,ColoradoCollege.

4. EARLY DISCUSSIONSOF LIMITS: GREGORY T. VINCENT,GALILEO,HOBBES.Limits n the Fifteenth nd SixteenthCenturies. Withthe fifteenthenturynew mathematical deas appear. These germsare found n Greek philosophy,but theyfailedto develop duringthe dark centuries. In the fifteenthenturythe German cardinal, Nicolaus Cusanus (1401-1465), considered variability

without being able to apply it successfully;he advanced the notionof a limit,thoughunable to pass correctly o the limit;he entertained he notionof nfini-tesimalsbut was not able to use them nan infinitesimalalculus.' He heldthatrules developed for the finite ose their validity for the infinite-a statementwhich aterthinkershave not always heeded sufficiently.A pointmovingwithinfinite elocity n a circle s each moment n every positionon the circle;henceit is at rest.Duringthe century, r century nd a half,after Cusanus, conceptsof imitsand processes nvolving hepassingto thelimitbeginto appear in differentartsof Europe, like flowers n a field n early spring. Perhaps first n time, n thedevelopmentof ideas consideredby Cusanus, is Giovanno B. Benedetti,a dis-tinguishedforerunner f Galileo, who broughtout a publication in 1585 atTurin, Italy. As early as 1586, and again in 1608, Simon Stevin at Leydenexhibited heprocessofpassingto thelimit.2 In 1604 the talian mathematician,Luc Valerio,publishedat Rome a treatise,De centrogravitatis,hich containsaremarkableapproach to the modern idea of limits.3 In Galileo's celebrated

1K. Lasswitz,GeschichteerAtomistik,amburg ndLeipzig, . Bd.,1890,pp.283,284,287,See alsoMax Simon, CusanusalsMathematiker, estschr.. Weber,eipzigundBerlin, 912.pp. 298-337.2 H. Bosmans, Sur quelques xemplese a m6thodees imiteshezSimon tevin, Annalesde a societecientifiquee Bruxelles, . 37, 1912-13, . fascicule.3H. Bosmans, Les d6monstrationsar l'analyse nfinitesimalehezLuc Valerio, Annalesde la Societe cientifiquee Bruxelles, . 37, 1912-13,2. fascicule;C. R. Wallner, Ueber dieEntstehungesGrenzbegriffes,ibliotheca athematica,. F., Bd. IV,.1903,p. 250,77

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78 ZENO S ARGUMENTS ON MOTIONdiscourses n mechanicsnd falling odies 1638) there re frequentnstances flimits. In the Netherlandsgain,Gregoryt. Vincent, hose esearches ave,untilrecently, ardly eceived herecognitionheydeserve,was familiar iththewritingsfLuc Valerio, ndhimselfontributedowardaying hefoundationsfor he nfinitesimalalculus. Similar tudies earing n the concept f a limitare due to Andreas acquet ofAntwerp,nd to JohnWallis n his Arithmeticainfinitorum,655,whowerebothfamiliar ith he Opusgeometricumf GregorySt. Vincent.'We proceednowto a specialmention f discussionsfZeno's arguments.Benedetti,whomwe mentionedbove, held that the flyingrrow, hought fat a point n its path, does not covera finite istance, ut it differsrom narrow t restby possessing he attribute f velocitywhich ersists ven n aninfinitesimalime and space.2 Direct referenceo Zeno in a mannerwhichexhibits eckless ollowingf hegreatdialecticians found n Giuseppe iancaniofBolognawho about 1615 sought o establish he ncommensurabilityf twolinesby theconsiderationhat supposed ommonmeasure ouldnotbe appliedto eitherine,because he measuremust irst e applied o half f t,andbeforethat ohalf f hathalf, nd so on toinfinity,hich s as impossiblen operationas Zeno's Dichotomy. 3Speculations fGalileo. Far moresuccessful han earlierwritersn theapplicationf nfinitesimalsereKeplerand Cavalieri, ut more mportantous at present re the speculations fGalileo. Galileo pproached he problemof nfiniteggregates ith keenness fvision nd an originalityhichwasnotequalledbefore he time of Dedekind nd GeorgCantor. Galileo's dialogueson mechanics,Discorsi e Dimostrazionimatematiche,638, opens the firstdaywith discussion f divisibilitynd continuityf matter nd space.4 Salviati,who n general epresentshe author's wn deas, says, 5 the nfinites incon-ceivable o us, as is the ast indivisible. Simplicio, ho n thesedialoguessthespokesman f Aristoteliancholastic hilosophy,emarks hat the infinityofpoints n a longerinemust egreater han he nfinityfpoints n a shorterone. Then come heremarkable ords fSalviati:

These difficultiesrise becausewe withour finitemindsdiscuss he nfinite,ttributingothe latterproperties erived rom he finite nd limited. This,however,s not ustifiable; ortheattributesreat, mall and equal are notapplicable o the nfinite,inceone cannot peakofgreater, maller, r equal infinities. . . If now ask how many quaresare there, ne cananswerwith ruth,ust as many s there reroots; or very quarehas a root,6 very oothas asquare,no square has more han one root,no rootmore han one square. I seenoescape,except o say: the totality f numberss infinite,he totality fsquares s infinite,hetotalityofroots s infinite;hemultitude f squares s not essthan he multitudefnumbers,eithers1C. R. Wallner, oc.cit.,p. 257.3 K. Lasswitz, p. cit.,Vol. II, p. 17.' J. C. Heilbronner,istoriamatheseosniversce,ipsiam,742,p. 175.4 See a German ranslationnOstwald's lassiker, o. 11, pp. 24-37, also No. 24, p. 17; anEnglish ranslation f theparts bearing n aggregatess givenby E. Kasnerin BulletinAm.Math. oc., Vol. XI, 1904-5,pp. 499-501.' Ostwald's lassiker, o. 11, p. 29.6 Following he custom fhis time,Galileoconsiders nly one root of a positivenumber,namely heprincipaloot.



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ZENO' ARGUMENTSON MOTION 79thelatter he greater; nd, finally,he attributes qual, greater,nd less are not applicable oinfinite ut solely o.finiteuantities.Weshall ee thatGalileohas been curiously isinterpretedy somewriters,includingauchy, s demonstratingere hat n actual nfinityas no existence.That there hould e as many quares s there re ntegersltogether as takenas absurd; hence the existence f actual infinity as considered isproved.Galileo's killntheuseof he nfinitendemonstrationss shown n thefollowingpassageon fallingodies:1

If the velocitywereproportionalo the distance hrough hichthas fallen r is to fall,then hosedistanceswouldbe passed over n equal times; hus, fthevelocitywithwhich bodyovercomes ouryards s to be doublethe velocitywithwhich he first woyardswere vercome,then hetimes eeded or hese woprocesses ouldbe the ame; but four ards an be overcomein the same time s two yardsonly n the case of nstantaneous otion;we see on thecontrarythat the body needstime o fall, nd that t needs esstimefor fall of two yards hanoffouryards;hence t is not true hat thevelocitiesncrease roportionallyothe distance allen.Gregoryt.Vincent. The most mportant iscussionfZeno given t thistime s that byGregoryt. Vincent,n hisOpus geometricumuadraturcirculietsectionumoni, ublished t Antwerpn1647,but writtenpparentlywenty-fiveyears earlier. It is a massivevolumeof 1400pages. Influencedn hisgeometrical esearches y the medieval cholastic onceptofthe continuum,accordingo which linedivided epeatedlysnotreducedo ndivisiblelementsas taught y theatomists,ut admits fbeing ubdivided d infinitum,regorySt.Vincentooka stepdifferentrom hatofArchimedes.While,nhisproofs,Archimedesept ndividing,nly ntil certain egree f mallness asreached,St. Vincent ermittedhe ubdivisionsocontinued nfinitum.Using nlimitedsectionngeometrye ntroducedgeometriceries hatwas rulyn nfiniteeries.2

I I i I I l 1A B C D E KThis much adbeen ccomplishedyat eastonewriter efore im,3 ut, o far snowknown, e sthefirsto apply he nfiniteeometricrogressiono the tudyof he Achilles. Taking definiteine egment K he divides t at B in a givenratio, hen he dividesBK in the same ratioat C, and so on. The segmentsAB, BC, CD, . . . form an infinite eometricprogression. The points C, D,E . . . lie,all ofthem,betweenA and K; theyapproachK as near as weplease,but (in accordancewithscholastic hilosophy) ever reach it. As Gregoryconceives hismatter, is an obstacle,o to speak, gainst he furtherdvanceofthe seriesofpointsA, B, C, . . . , similarto a rigidwall. Terminus pro-gressionisst serieifinis,d quemnullaprogressioertinget,icet n infinitumcontinuetur;ed quovis nteruallo atopropriusd eum accedereoterit. Byseries s meant he egment K, by progressio, he egments B, AC, . .

IOstwald'sKlassiker, o. 24, p. 17.2 Gregoryt. Vincent, pusgeometricum,. 1, pp. 51-56,95-97; for ur knowledgef thispartofthebookwe are dependentntirelyponC. R. Wailner's ccountn theBibliotheca athe-matica, . F., Vol. IV, 1903,pp. 251-255.aSee H. WieleitnernBibliotheca athematica,. F., Vol. 14, 1914,pp. 150-168.



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80 ZENO S ARGUMENTS ON MOTIONGregory tates his conclusion hus: Dico magnitudinem K aequalem esse totiprogressionimagnitudinum ontinue proportionalium, ationis AB ad BC ininfinitumontinuatae;siue quod idem est, rationisAB ad BC in infinitumon-tinuatae terminum sse K. Considering he Achilles in this connection,he associates thisparadoxon motionforthe first imedefinitely ith the sum-mation of an infiniteeries. Moreover,Gregory t. Vincent s the firstwriterknownto us who states the exact time and place of overtaking he tortoise.So far as we are able to ascertain,Gregorywas not troubled, n explaining heAchilles, by thefactthat in his theory, he variabledoes not reach ts limit.Nor, apparently, id this matter roublehis readers. His mode of solvingtheproblem ppealed to many. We shall see that Leibniz makes special referenceto it. Over a century fter Gregory'spublication,Saverienrefers n his dic-tionary' o the Achilles, dont Gregoirede Saint Vincenta faitvoirla faus-sete. Formeygave Gregory t. Vincent'sexplanation n the article Mouve-ment in Diderot's Encyclopedie 1754), later reprinted n the Encyclopediemethodique, nd in 1800 translatedat Padova into the Italian language. Thedefinitionf a limitas givenin the Encyclopediemethodiqueoes not allow thevariable to surpass ts limitbutplaces no obstacle n theway of itsreaching tslimit.Descartes, De Morgan and Others. Descartes at one time discussedtheAchilles. His treatment s much like that of GregorySt. Vincent. It isgiven n a letterofJuly,1646,to Clerselier.2 He letsAchilles, r in his place ahorse,be, at the start,10 leaguesbehind hetortoise, utmoving entimesmorerapidly hanthe latter. The realdifficultyf theparadoxhedoes nottouch,forhe says:

L'AchilledeZenonne serapasdifficile oudre,i onprend arde ue,siA a dixieme artiede quelquequantit6 n adioute a dixi6me e cettedixi6me,ui estunecenti6me, encore adixi6me e cette erniere,ui n'estqu'unemilliesmee la premiere, ainsiA 'infini,outes esdixi6mesointes nsemble,uoyqu'elles oient upos6es 6ellementnfinies,ecomposentoutes-fois u' unequantit6 inie,qauvoir ne neusi6mee la premiereuantit6 . . Et la caption sten ce qu'on magineue cetteneusi6meartie 'une ieueestune quantit6nfinie, causequ'onla diviseparson maginationn despartiesnfinies.Descartes lookeduponthe actually nfinites mysterious,ut not impossibleor absurd. He seemedto accept it in the abstract,but deny t in the concrete.At this time and even earlier see the foregoingxtracts romGalileo) therewastalk about the finitude f the humanmindand its consequent nability o con-ceive the infinite. This was ridiculedby De Morgan. He claimedthat ifthehumanmind s limited,we tacitlypostulatethe unknowable ; moreover, venifthehumanmindwerefinite,here s no morereasonagainst ts conceiving heinfinite hanthere s for mindto be blue in order o conceiveofa pair ofblueeyes. Or, as De Morgan puts it in anotherplace,- he argument mounts to

this, who drivesfatoxen shouldhimself e fat. FromDescartesto Hamilton,I Saverien, ictionnaireniverselemathematiquet de physique,aris,1753,Art. Mouve-ment.2 OeuvreseDescartesar CharlesAdametPaul Tannery, . IV, pp. 445-447.



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ZENO S ARGUMENTS ON MOTION 81says De Morgan,' this doctrine s accepted by manyminds. But its genesis sfound, s we have stated, longbeforeDescartes.A whollydifferent,ut no more satisfactoryxplanationof Achilles comesfrom anotherFrenchman of that time, Pierre Gassendi, the physicist. In hisview Zeno's proofsneed no refutation,f with Epicurusone assumesnot pointsbut atoms. A difficultyeems to arise fromdifferencesn velocityof motion,for n thesame time that a body moves overthephysicallyndivisible, he morerapid body must travel over several indivisibles. In his opinionthis difficultymay perhaps be overcome by conceivingmotionas discontinuous, nd slowermotionas a mixture frestand motion. To the senses motionwould still seemcontinuous.2 To those who experienceddifficultiesn acceptingthe existenceof ndivisible toms, he capuchin,Casimirof Toulouse. offers n easysolutionbyremindinghat angelshad extension,yetwerephysicallyndivisible.3It is worthyof note that John Dee, the famous astrologerwho wrote anelaborate mathematicalprefaceto Billingsley's dition of Euclid (1570), departsfrom he contention hat two lines containingthe same number of parts mustbe ofequal length. He says:

Our leastMagnitudesan be divided nto omanypartes s thegreatest. As, Lineof ninch ong with s) maybedivided nto s many artes, s maythediameter f thewholeworld,from ast to West:orany wayextended.Discussion of Thomas Hobbes. The earliest Britishwriter, fter DunsScotus,to take up explicitly eno's argumentss thephilosopher, homasHobbes(1588-1679). In 1655he wrote:4

theforce f thatfamous rgumentf Zenoagainstmotion, onsistedn thispropo-sition, hatsoever aybe divided ntoparts, nfiniten number, he ame is infinite;which ewithoutdoubt,thought o be true,yetneverthelesss false. For to be divided nto nfinitearts, snothinglsebutto be dividednto s many arts s anymanwill. But it snotnecessaryhatline houldhaveparts nfinitennumber,rbe infinite,ecause candivide ndsubdivide t asoften s I please;forhowmanyparts oever make,yettheir umbers finite; ecausehe thatsays parts, imply,withoutddinghowmany,does not imit ny number, utleaves t to thedeterminationfthehearer, herefore esay commonly, linemaybe dividednfinitely; hichcannot e true nany other ense.With-Hobbes, infinites synonymouswith ndefinite.He takes an agnosticattitudetoward problemsof nfinity:But whenno more s said than his, umber s infinite,t is to be understoods if t weresaid,thisnamenumber s an indefinite ame. . . And,therefore,hatwhich s commonlyaid,thatspaceand timemaybe divided nfinitely,s notto be so understood,s iftheremight eany nfiniter eternal ivision;butrather o be taken nthis ense,whatevers divided s dividedinto such parts as may again be divided. . . . Who can commend imthatdemonstrateshus?'If the worldbe eternal, henan infinite umber fdays,or othermeasures f time, recededthebirth fAbraham. But thebirth fAbraham receded hebirth f saac; andthereforene

IA. De Morgan, On Infinity;nd on theSignofEquality, n Trans.ofthe CambridgePhilosoph. ociety, ol. XI, p. 157,Cambridge,871 readMay 16, 1864].2 Gassendi,Operaomnia, 658, , p. 300a. An. I, p. 239; Lasswitz, p. cit.,Vol. II, p. 150.3 Lasswitz, p. cit.,Vol. II, p. 494.4The EnglishWorks fThomasHobbes, ol. I, London, 839,pp. 63, 64, 413. Hobbes refersto Zeno'sargumentslso inhis Latinworks. See ThomaeHobbes,Operaphilosophica,ol.V,Londini, 845,pp. 207-213.



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82 A GENERAL FORMULA FOR THE VALUATION OF SECURITIESinfinites greater hananother nfinite,r one eternal han another ternal; which,'he says,'is absurd.' Thisdemonstrations likehis,whofrom his, hatthe number f evennumberssinfinite, ouldconclude hatthere re as many vennumberss there re numbers imply, hatis to say, theevennumbers re as many s all the evenand oddtogether. Theywhich nthismanner akeaway eternityrom heworld, o theynotbythe same means akeaway eternityfrom he creator ftheworld? . . And thementhatreasonthusabsurdly re not diots, ut,whichmakesthisabsurdity npardonable, eometri as, and such as take uponthemto bejudges, mpertinent,utsevereudgesof othermen'sdemonstrations.

The reference o odd and even numbers doubtless arose from his contactwithGalilean thought. Whilesojourningon the Continent,he had gone to seeGalileo,thena prisoner. Hobbes thoughthe had effected he duplicationof thecube and the squaringof the circle. On this matterhe became involved in aheated controversywiththe algebraist,JohnWallis. The aged Hobbes was nomatch gainst youngWallisonmathematical uestions. When the mathematicalworksofWalliswerebeing brought ut,Wallis refused o allow his controversialmatter against Hobbes to be incorporated n them.' Whether the whole isgreaterthan a part was an issue touched upon duringthis dispute. Hobbessaid to Wallis: All thisarguing f nfinitiess but the ambitionof schoolboys.It cannot be said that Hobbes made any real contribution o a deeper under-standingof the Achilles or any of Zeno's otherargumentson motion. Hisobjectionto thedictum, whatevermaybe divided ntopartsinfinitennumber,the same is infinite, s no new contribution;Aristotlehad advanced that far.How Achilles caught the turtle is beyond comprehension hrough our sensualimagination;Hobbes nowhereexplainsthis inability. However,he does touchupontheconceptof a limit n his controversy ithWallis. Hobbes charged hatsome oftheprinciples ftheprofessorsre voidofsense ; oneof thoseprinciplesbeing, that a quantity may growless and less eternally, o as at last to beequal to anotherquantity; or,which s all one,that there s a last in eternity. 2

A GENERAL FORMULA FOR THE VALUATION OF SECURITIES.3By JAMES W. GLOVER, University f Michigan.

The object of this paper is to derivea formulafor the valuation of a verygeneral type of securities. The security s redeemed n r equal installments tintervalsoft years,the first edemption eingmade after years. The annualrate ofdividend s g payable in m installments,nd the securitys purchasedtorealize the investor nominal rate ofinterest withfrequency f conversionm.1A full ccount f he controversyetween obbes nd Wallis s givennCroomRobertson'sHobbes, p. 167-185.2 TheEnglishWorks f ThomasHobbes, ol. 7, p. 186.3Read before he Chicago Sectionof the AmericanMathematical ociety,April,1912.Those unfamiliar ith henotation nd functionsmployedn the theory fcompound nterestmayconsult ext-Bookf he nstitutefActuaries, art , by RalphTodhunter;TheMathematicalTheory f nvestment,yErnestB. Skinner;Bulletin ftheDepartmentf Agriculture,o. 136,on Highway onds,by Laurence . Hewes and JamesW. Glover.

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Cajori - The History of Zeno's Arguments on Motion - Phases in the Development of the Theory of Limits