On the Homocentric Spheres of Eudoxus Author(S): Ido Yavetz Source: Archive for History of Exact Sciences, Vol

On the Homocentric Spheres of Eudoxus Author(s): Ido Yavetz Source: Archive for History of Exact Sciences, Vol. 52, No. 3 (February 1998), pp. 221-278 Published by: Springer Stable URL: http://www.jstor.org/stable/41134047 . Accessed: 26/09/2014 22:57

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On theHomocentric Spheres ofEudoxus

Ido Yavetz

Communicatedby N. Swerdlow

"In tale dubbioio seguiròl'usato metodo, di non decidereintorno a ciò che più nonè possibilesapere;" G.V. Schiaparelli,1877.

a. Introduction

In 1877,Schiaparelli published a classic essay on thehomocentric spheres of Eu- doxus.In theyears that followed, it became the standard, definitive historical reconstructionof Eudoxian planetary theory. The purpose of this paper is to showthat the two texts on whichSchiaparelli based his reconstructiondo notlead in an unequivocalway to thisinterpretation, and that they actually accommodate alternative and equally plausible interpretationsthat possess a clearastronomical superiority compared to Schiaparelli's. One shouldnot mistake all ofthis for a call toreject Schiaparelli's interpretation in favor ofthe new one. In particular,the alternative interpretation does notrecommend itself as a historicallymore plausible basis for reconstructing Eudoxus's and Callippus's planetary theoriesmerely because of itsastronomical advantages. It does, however,suggest that theexclusivity traditionally awarded to Schiaparelli'sreconstruction can no longerbe maintained,and thatthe little historical evidence we do possess does notenable us to makea justifiablechoice between the available alternatives.

b. Schiaparelli'sReconstruction of theEudoxian System

Our currentknowledge of Eudoxus's homocentricspheres relies primarily upon two sources.His own writingshave not survived.1The earliestknown reference to Eudoxus'sastronomical work is a shortpassage in Aristotle'sMetaphysics. Coming froma contemporaryofEudoxus, and one who is generallyregarded as ourmost reliable sourceon ancientGreek thought, the importance of this reference is clear.Here is how Aristotledescribed Eudoxus's arrangement:

1 For a surveyof thesources, ancient and modern,on thehomocentric system devised by Eudoxus,see E. Maula,"Eudoxus Encircled," Ajatus, 33 (1971): 201-253, esp,pp. 206-211.

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Eudoxusassumed that the sun and moon are moved by three spheres in each case; thefirst of these is thatof the fixed stars, the second moves about the circle which passesthrough the middle of the signs of the zodiac, while the third moves about a circlelatitudinally inclined to thezodiac circle; and, of the oblique circles, that in whichthe moon moves has a greaterlatitudinal inclination than that in which thesun moves. The planetsare moved by four spheres in each case; thefirst and secondof theseare thesame as forthe sun and moon,the first being the sphere of thefixed stars which carries all thespheres with it, and thesecond, next in orderto it,being the sphere about the circle through the middle of thesigns of thezodiac whichis commonto all theplanets; the third is, in all cases, a sphere withits poles on thecircle through the middle of the signs;the fourth moves abouta circleinclined to themiddle circle (the equator) of the third sphere; the poles of thethird sphere are differentfor all theplanets except Aphrodite and Hermes,but for these two the poles are thesame2

Concentratingfirst on the theory of the five planets, it is notdifficult to understandthe role ofthe first two spheres in Aristotle's description. Obviously, the first - henceforth "stellar sphere"- accountsfor the diurnal motion common to all theheavenly components. The second- henceforth"ecliptic sphere" - createsa componentof motionconfined to the plane of theecliptic for the five planets as well as forthe sun and moon.As forthe thirdand fourthspheres, Aristotle's text is less helpful.All he relatesabout them is that thepoles of thethird rest on diametricallyopposed points of theecliptic, and thatthe poles of thefourth are fixedinside the third at an unspecifiedinclination to thethird sphere'spoles. Schiaparelli'sclassical reconstructionof the Eudoxianscheme is not deriveddirectly from Aristotle's Fourth Century BC passage,but from the commentary of Simplicius,written about 900 yearslater.3 Simplicius was morespecific about the particularfunction of the third and fourthspheres: And,as Eudemusrelated in thesecond book of his astronomicalhistory, and Sosigenesalso whoherein drew upon Eudemus, Eudoxus of Cnidos was thefirst of theGreeks to concernhimself with hypotheses of thissort, Plato having, as Sosigenessays, set it as a problemto all earneststudents of thissubject to find whatare theuniform and orderedmovements by theassumption of whichthe phenomenain relationto themovements of theplanets can be saved. . . . The thirdsphere, which has itspoles on thegreat circle of the second sphere passing throughthe middle of the signsof the zodiac, and whichturns from south to northand fromnorth to south,will carryround with it thefourth sphere which also has theplanet attached to it,and willmoreover be thecause ofthe planet's movementin latitude.But notthe third sphere only; for, so faras it was on the

2 Aristotle,Metaphysics, 1073b 17 - 1074a 15, in Sir ThomasL. Heath,Greek Astronomy, (London:J.M. Dent & Sons LTD., 1932),pp. 65-66. 3 Simpliciusbased his accountmainly on theSecond CenturyAD writerSosigenes, whose ownwork is notbased directlyon Eudoxus,but on Eudemus,Aristotle's student who flourished duringthe Fourth Century BC. The worksof bothSosigenes and Eudemushave been lost(see Maula,note 1, p. 210).

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thirdsphere (by itself),the planet would actually have arrivedat the poles of the zodiac circle, and would have come near to the poles of the universe;but, as thingsare, the fourthsphere, which turnsabout the poles of the inclined circle carryingthe planet and rotatesin the opposite sense to the third,i.e. fromeast to west, but in the same period, will preventany considerable divergence(on the partof the planet) fromthe zodiac circle, and will cause the planet to describe about this same zodiac circle the curve called by Eudoxus the hippopede(horse- fetter),so thatthe breadthof the curve will be the (maximum) amount of the apparentdeviation of the planet in latitude,a view forwhich Eudoxus has been attacked.4 The geometricalarrangement of spheres 2, 3, and 4 that Simplicius described has been interpretedby Schiaparelli as illustratedin Fig. 1 below. The two inner spheres rotateat equal speeds in opposite directions,so thatthe planet's motion is the super- position of the two rotations.The trace projected by the moving planet on the surface of the ecliptic sphere (and on the stellar sphere, as long as the ecliptic sphere remains stationary)is the hippopede, literallymeaning horse-fetter,shown in Fig. 2. Its long axis of symmetrycoincides with the ecliptic - the vertical great circles in Figs. 1 and 2. The hippopede's projection on the horizontal equatorial plane of the ecliptic sphere (perpendicularto the plane of the paper) is a circle. Consequently,the hippopede is also the intersectionof the ecliptic sphere with a cylinder whose base diameteris equal to the differencebetween the radius of the ecliptic sphere and the radius of eitherof the boundinglatitude circles at the top and bottomof the hippopede. The central angle of these latitudes measured from the equatorial plane is equal to the inclinationbetween the axes of spheres3 and 4, which is also equal to the angle at

Figure1

4 Simplicius,In Aristotelisde Cáelo Commentario,edited by I.L. Heiberg,(1894), pp. 488: 18-24;496: 23^497:6, trans,by Heath, Greek Astronomy, pp. 67-68.

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Figure2

Figure3 whichthe hippopede intersects itself. The inclinationbetween the axes ofspheres 3 and 4 (set to 35°in Fig. 2) is thereforethe single variable parameter that fully determines theshape of thehippopede - thelarger the angle, the longer and widerthe hippopede. Combinationof the hippopedal motion generated by the rotations of spheres 3 and4 with

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a simultaneousrotation of theecliptic sphere, projects on thestellar sphere (not shown in Fig. 1) themotion of theplanet relative to thefixed stars. The traceof thismotion circuitsthe stellar sphere while wobbling slightly above and below the projection of the ecliptic,which marks the center of the zodiacal belt. Most importantly, ifthe speeds of spheres3 and 4 are properlychosen, the general circuital motion of theplanet around theecliptic is interruptedat fixed intervals by retrograde motion. An exampleof thisis shownin Fig. 3, wherespheres 3 and4 thatcreate the hippopede of Fig. 2 are madeto revolveexactly three times as fastas theecliptic sphere. The superpositionof the diurnal rotation of the stellar sphere over this motion gen- eratesthe full motion of a planetrelative to the horizon of the earthbound observer. Each of theplanets in thisscheme has its own set of fourspheres, the inner two of which createa hippopede.The differencesbetween the resulting planetary paths stem from the differentsize ofthe hippopede that pertains to each ofthem and fromthe ratio between therotation period of spheres3 and 4 and therotation period of theecliptic sphere of each planet.Thus did Schiaparelliturn the hippopede into the retrogression generator, and thecore of the Eudoxian planetary theory.

c. An AristotelianAlternative, and a Restrictionby Simplicius

Thereexists, however, another way of interpreting the texts of Aristotle and Simpli- cius.This alternative will be arrivedat here by first reading Aristotle's text as a blueprint forthe construction of a generalgeometrical model, which will thenbe furthercon- strainedby referenceto Simplicius'stext. Aristotle explicitly specified the following detailsabout the four spheres of the Eudoxian scheme: 1. The firstsphere is thesphere of the fixed stars, and its motion is impartedto all three remainingspheres. Note that Aristotle's text does not say that this sphere rotates with theperiod of thestars' diurnal motion. We ascribeto it thediurnal period because thismakes the system work properly. 2. The secondsphere rotates in theplane of the ecliptic. Here, too, Aristotle refrained fromspecifying its speed,and we ascribeto it themean eclipticperiod for each particularplanet to makethe system work properly. This point will become particu- larlyrelevant later, upon examination of Simplicius'sproblematic discussion of the speedsof the second and third spheres in thetheory of the moon. 3. Thepoles of the third sphere are fixed to diametrically opposed points on theecliptic. 4. The poles of thefourth sphere are inclinedby an unspecifiedangle relative to the poles ofthe third sphere. 5. As withthe first two spheres, Aristotle wrote nothing about the speeds of rotation of thetwo inner spheres. Here, as in Schiaparelli'sinterpretation, the assignment of specificspeeds is guidedby the assumption that the system's primary purpose was to accountfor planetary retrograde motion in terms of uniform homocentric rotations.5

5 BernardGoldstein has recently observed that in the accounts of astronomical knowledge fromthe 4th, 3rd, and 2nd centuries BC, thereis no termcorresponding to the words "retrogrademotion." (B.R. Goldstein, "Saving the Phenomena: The Background toPtolemy's Planetary

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6. Finally,nowhere in Aristotle'stext is thereany indication that the planet must be affixedto theequator of the fourth sphere, nor is anysuch requirement necessitated bythe basic programof reducing the motion of the heavens to uniformhomocentric rotations.Indeed, the majority of stars on thestellar sphere are placed away from its equator.

Keepingthese points in mind,the arrangement depicted in Fig. 4 belowis just as consistentwith Aristotle's text as Schiaparelli's.Note that this arrangement possesses two variableinclination parameters as opposedto the single one that characterizes Schiapar- elli's interpretation,namely, the inclination between the axes ofspheres 3 and4, andthe inclination,or latitude,of the planet relative to thepole of sphere4.

Figure4

Theory"Journal for theHistory of Astronomy, 28 (1997): 1-12, esp. pp. 3-4). However,as a workinghypothesis, the assumption that Eudoxus and Callippus designed their systems primarily to accountfor retrograde motion provides a naturalexplanation for the associationof fouror fivespheres with each individualplanet, and Goldsteinsuggested no plausiblealternative reason fortheir existence. One could conjecturethat since alreadyPlato knewthat both Mercury and Venusmove sometimesfaster and sometimesslower than the sun, the purpose of theEudoxian arrangementwas togive each planet a variablespeed. This, however, can be crudelydone with just threespheres per planet, and makesCallippus's use of fivespheres for each of Mars,Venus, and Mercuryquite mystifying. Furthermore, aside fromAristotle's short, general account, practically nothinghas remainedof theoriginal treatises on thesubject of planetarymotion from the time of Eudoxus,Callippus, and theirimmediate successors. This weakensthe impact of Goldstein's philologicalargument from silence, although it does notwarrant the rejection of thepossibility thatindeed, retrograde planetary motion was notrecognized by Eudoxus and hiscontemporaries, and thatthe Eudoxian homocentric system was meantto addressentirely different concerns. If anything,this further emphasizes the need for a generalsuspension of judgement along side with thedevelopment of severalpossible historical reconstructions.

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This arrangementmay be anachronisticallydescribed as a sphericalversion of theco- planardeferent and epicycle arrangement that became the staple of Ptolemy's planetary theory.When the two inner spheres spin at equal speeds(relative to theecliptic sphere) in oppositedirections, the planet traces a closedloop (qualitativelyillustrated in Fig. 5). Unlikethe plane version,however, the spherical counterpart is sensitiveto switching thesizes of thedeferent and theepicycle as shownin Figs. 6 and 7. In otherwords, thenon-commutative nature of thetwo inclinations actually increases the flexibility of thisarrangement. In additionto thetwo inclinations, this arrangement can makegood use of a thirddegree of freedombecause the initial inclination of theaxis of sphere3 relativeto theecliptic sphere's axis need notbe confinedto theplane of theecliptic. This definesa phase angle (Fig. 8), whichorients the loop's long axis of symmetry alonga greatcircle other than the ecliptic, while keeping the loop's centerof symmetry at thepoint on theecliptic where the pole of sphere3 is attached(Fig. 9).6 This loop replacesthe hippopede of Schiaparelli's interpretation as the retrogression generator. As Fig. 10 shows,the combination of motion along this loop withecliptic rotation generates planetarytraces around the ecliptic that are markedly different from those generated by thehippopede.

/ ' ^'

Figure5

6 Similarly,the initial position of the planet need not be confinedto thebottom of the latitude circledrawn around the pole ofsphere 4. Thereare actually two degrees of freedom here, but they aredegenerate to theextent that both rotate the resulting loop's axis of symmetryrelative to the projectionof theecliptic. Because sphere4 rotatestwice as fastrelative to sphere3 as thelatter rotatesrelative to the ecliptic sphere (sphere 2), thephase angle of the fourth sphere must be twice thephase angle of the third sphere to effect the same rotation of the loop's axis ofsymmetry relative to theecliptic. The startingposition of theplanet on theloop willnot, in general,be thesame in bothcases, but for the purpose of investigating the trace of retrogression this is inconsequential.

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Figure6

Figure7

The loop in Fig. 6 is createdwhen the axis of sphere4 is inclinedby 19° relativeto the axis of sphere3, and theplanet is on latitude16° measuredfrom the pole of sphere4. In Fig. 7 the twoaxes are inclinedby 16°,while the planet rests on latitude19° of sphere4

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Figure8

Figure9

The phase angle determinedby the startingposition of the planet is also available in Schiaparelli's arrangementwith the same effect,namely, it orientsthe hippopede's long axis of symmetryalong a greatcircle thatis inclined to the ecliptic, and intersects it at the poles of the thirdsphere. However, in marked distinctionto the alternative retrogressiongenerator, whose centerof symmetrycoincides withthe pole of sphere 3,

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Figure10 The loop in Fig. 9 is generatedby the same inclinationsthat create the loop in Fig. 6, butlies alonga greatcircle that intersects the ecliptic at a phaseangle of 5°. Figure10 showsthe traceprojected onto the stellar sphere by motion along the loop ofFig. 9 witha periodthree times shorterthan the rotational period of the ecliptic sphere the hippopede's centerof symmetrylies on the equator of sphere 3, 90° of latitudeaway fromeither pole. In Schiaparelli's arrangement,therefore, starting the planet fromthe intersectionof the equators of spheres 3 and 4 (Fig. 1), but with a phase angle other than 90°results in moving the hippopede's centerof symmetryaway fromthe ecliptic. Consequently,the entireplanetary trace would be displaced to one side of the ecliptic, instead of creating an equivalent of the ecliptically balanced trace generated by the skewed loop of the alternativearrangement (Fig. 10). Small phase angles would shiftthe entirepath of the planet above or below the ecliptic withoutappreciable change in the formof the retrogradephases, while large phase angles would completelyremove the planet's course fromthe zodiacal belt, which is clearly unacceptable. Note, therefore, thatanother tacit assumption is made in Schiaparelli's interpretation,namely, that the pole of the fourthsphere is inclined relativeto the pole of the thirdalong a greatcircle thatintersects the projection of the ecliptic on sphere 3 at an angle of 90°. Any other orientationfor the directionof inclinationwill move the centerof the hippopede away fromthe ecliptic and will resultin impropermotion. We have seen, then,that Aristotle's account of the Eudoxian systemadmits an alternativeinterpretation that satisfies its explicit specificationsas fullyas Schiaparelli's interpretation.The alternativeinterpretation possesses significantlyincreased flexibility,generates planetarymotion thatis markedlydifferent from the motion created by Schiaparelli's interpretation,and thus farmakes no use of a retrogressiongenerator that

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions On theHomocentric Spheres of Eudoxus 23 1 can be described as a hippopede - a word thatAristotle does not mention.It also takes no account at all of Simplicius's commentaryon Aristotle'stext, to which we turnnext. The firstpoint of interestin Simplicius's account is the observationon the planet's motionwithout a fourthsphere. Heath's translationhas Simplicius sayingthat with three spheresonly, the planet would arriveat thepoles of theZodiac. Simplicius's actual words, however,could as plausibly be translatedas saying thatthe planet would move toward thepoles of the Zodiac.7 It is not clear fromthe textwhether it indicates thatthe planet would always arriveat the poles of the Zodiac, or be displaced in theirdirection without necessarilyarriving at themalways.8 The second possibility,which includes the firstas an extremecase, is clarifiedin termsof the alternativeinterpretation as follows. It is, in fact,possible to create retrogrademotion without a fourthsphere, by placing the planet on the thirdsphere at the point where the pole of the fourthsphere is attached.As the rotatingthird sphere is carriedby therevolving ecliptic sphere,the combined motionwill projecta spiral path on the stellarsphere, providing that the thirdsphere rotateswith a sufficientlyshorter period than the ecliptic sphere. This means manyretrogressions in the course of a single ecliptic revolution.If we wish to reduce the numberof retrogressions per ecliptic period (an acute need in the cases of Mars and Venus) we need to increase therotational period of the thirdsphere; but then we mustcompensate forthe speed lost by the increased period of revolution.To do thatwe move the planet furtheraway from the pole of the thirdsphere, or towardthe poles of the ecliptic, to enlarge the circle it describes. The fastestmotion that the planet can reach fora given period of revolution is always obtainedby placing theplanet on theequator of the thirdsphere. But since the thirdsphere's pole restson theecliptic, the equator of thethird sphere necessarily passes throughthe poles of the ecliptic, and this would bringthe planet rightto the poles of the ecliptic. In short,the gist of Simplicius's observationis thatwithout a fourthsphere the arrangementwould create unacceptable departuresfrom the Zodiacal belt toward the Zodiacal poles, while actual planetarymotion never displays such departures.The passage is explained in a similar mannerin the contextof Schiaparelli's interpretation wherethe planet is always placed on the equator of the fourthsphere. Presumably,that is whereit would be placed on the thirdsphere were the fourthsphere to be eliminated, with the outcome of carryingthe planet all the way to the poles of the ecliptic. In both interpretationsthe fourthsphere constrainsthe latitudinalmotion of the planet, preventingit fromsuch unacceptable deviations.So far,then, neither interpretation can be plausibly ruled out on the basis of Simplicius's text. A second pointof interestinvolves Simplicius's explicitdiscussion of the rotational motionof the thirdand fourthspheres. Unlike Aristotle,who merelysaid thatthe two spheresrotate in inclined planes, Simplicius specifiedthat they do so at equal speeds,

7 Simplicius,InAristotelis de Cáelo Commentario,edited by I.L. Heiberg,( 1 894), pp. 496:23- 497:6. Schiaparellitranslated this passage as sayingthat with the third sphere alone theplanet wouldadvance toward the pole ofthe Zodiac ("Le SfereOmocentriche di Eudosso,di Callippoe di Aristotele,"in ScrittiSulla Storiadella Astronomia Antica, 3 vols.,[Bologna: Nicola Zanchelli, Editore],voi. 2, p. 100). 6 I thankAnthony Grafton and Reviel Netz for their help concerning the meaning and possible translationsof Simplicius'spassage.

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butin opposite directions. Unfortunately, rather than clarifying Aristotle's passage, Sim- pliciusmainly succeeded in beingambiguous, regarding both the direction and speed ofrotation. Regarding first the direction, only when two rotations take place aboutpar- allel axes can theymeaningfully be describedas coincidentor opposed,but the axes of thethird and fourthspheres are generallyinclined with respect to each otherin the Eudoxiansystem. The standardway of getting round this difficulty has beento assume thatSimplicius tried to implythat as longas thecoincident rotational components are largerthan the opposed ones, the rotations are to be consideredin thesame directions, and once theopposed components become dominant, the rotations are to be considered opposed.As a result,it has come to be acceptedthat the inclination angle between the axes of spheres3 and 4 cannotexceed 90°, despitethe fact that neither Simplicius nor Aristotlemade this stipulation explicitly. The pointis actuallyof no greatconsequence, becausesuch angles occur only in thetheory of Venus, whose full retrograde loop cannotbe observeddirectly against an immediatebackground of fixedstars because of its proximityto thesun.9 However, as a matterof principlethere is another,more natural wayof resolving Simplicius's ambiguous observation without the additional restriction on inclinationangles. Rotations always take place abouta pole thatmay be markedas "north"for the purpose of discussion,and theirdirections are bestdescribed relative to thispole, eitherwestward, or eastward.Simplicius might have been tryingto say that relativeto theirrespective north poles therotations of spheres3 and 4 wouldalways be opposed.Note that with this definition, ifthe two north poles are pointed180°from each other,the sphereswill appearto rotatein the same directionto an outsideob- server,but relative to theirown poles theyalways remain opposed - one clockwise,the othercounterclockwise. This interpretationofthe passage is freeof theambiguities of thetraditional way of understandingSimplicius's observation, and has theadditional advantageof notimposing on thetheory an additionalrestriction that is notindicated anywherein thetexts of Aristotleand Simplicius.10 Of greaterconsequence is Simplicius'sstatement that the spheres revolve with equal periods.Here again, the text leaves room for more than one interpretation.In Schiapar- elli's interpretation,Simplicius's reference to equal periodsin oppositedirections does not describethe apparentmotion of the two innerspheres, but provides a two-stage procedureby whichto setthe system in motion.First, spin the inner sphere. Then spin theouter one in theopposite direction but with the same period as theinner sphere and combinethe two motions according to therule that all rotationsare communicatedin-

9 Venusitself, as opposedto itsimmediate stellar background, can be observedduring nearly all of itsretrograde phase. Therefore, given a good chronometer,a precise global mapping of the fixedstars on thestellar sphere, and a sufficientlyaccurate data base ofstellar risings and settings, it is possibleto reconstructthe form of Venus'sretrograde path indirectly. However, as we shall see lateron, it appearsunlikely that in Eudoxus'stime the Greeks possessed either a sufficiently accurateglobal star map or a sufficientlyaccurate table of risings and settings. 1U Alternatively,one maysee thisin termsof proceduralinstructions that require none of the modernterminology for describing rotations: first the two poles must be aligned,then the spheres mustbe spunin opposite directions. Having done that, the poles of the oppositely spinning spheres maybe inclinedto anydesired angle. This procedureis exactlyequivalent to theinterpretation givenin thetext above.

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions On theHomocentric Spheres of Eudoxus 233 ward. This will produce the motionthat generates Schiaparelli's hippopede, but it must be noted thatSimplicius's textdoes not allude to any such procedure.Instead, it simply describes a systemof two homocentricspheres, rotating at equal periods in opposite directions.This does not supportSchiaparelli's reconstruction,where the inner sphere appears to stand still when its pole is aligned with the pole of the fourthsphere, or merelyto wobble back and forthwhile its inclinedpole orbitsabout thepole of the third sphere.For thetwo spheresto appear to rotatewith equal periods in opposite directions as Simplicius's textindicates, one needs firstto spin theinner sphere, and thento spin the outerone in the opposite directionat twice theperiod initiallygiven to the innersphere. As the two motionscombine underthe rule thatall rotationsare communicatedinward, the speed of the innersphere is reduced by one half,and the two spheres will appear to rotatein opposite directionswith the same period,as Simplicius's textrequires. With no furtherhelp fromSimplicius, it is quite naturalto take thetext at face value as alludingto the actual appearance of two concentricspheres, rotating with equal periods in opposite directions.Such a motion will not generatethe hippopede of Schiaparelli's reconstruction,but the loops of the alternativeinterpretation of Aristotle'saccount. Simplicius did not provide additional informationwith which one can make a sound choice between these two possibilities. Thus, the two alternativesthat fitAristotle's account because of its generality,also accommodate Simplicius's account by reason of its ambiguous nature. Finally,Simplicius explicitlywrote that the combined motionof thethird and fourth spheresproduces what Eudoxus called the"hippopede." Nowhere did Simplicius identify this figurewith the intersectionof a sphere and a cylinder,or a sphere with two cones joined at theapex. Proclus,however, used theterm to describe one of theso called "spiric sections" in his commentaryon Euclid:

Of the spiric sections one is interlacedlike a horse's hobble, [innov Ttéôrj] anotheris broad in themiddle and thinsout at the sides, and anotheris elongated 1 and has a narrowmiddle portionbut broadens out at the two ends.1

A littlelater, Proclus commentedon the originof these curves:

... we have at once a notion of the most elementarykinds of surfaces,the plane and the spherical,though it is only throughscience and scientificreasoning that we discover the varietyof surfaces that arise by mixture.What is remarkable about themis thatfrom the circle therecan oftenbe generateda mixed surface. This is whatwe say happens in thecase of thespiric surfaces, for they are thought of as generatedby therevolution of a circle standingupright and turningabout a fixedpoint that is not the centerof the circle. Thus threekinds of spiric surface are generated,for the center lies eitheron thecircumference, or inside, or outside the circle. If the centeris on the circumference,the continuousspiric surfaceis generated;if withinthe circle, the interlacedspiric surface,and if outside, the

11 Proclus,A Commentaryon theFirst Book of Euclid's Elements,Translated by GlennR. Morrow,(Princeton: Princeton University Press, 1992), p. 91 (112.5-112.8). See also, Heath, Euclid'sElements, Book I, (New York:Dover Publications, Inc., 1956),pp. 162- 163.

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open spiricsurface. And thespiric sections are threein numbercorresponding to thesedifferent kinds of surface.12 On thestrength of these passages, Heath declared: Thereis no doubtthat Schiaparelli has restored,in his 'sphericallemniscate', thehippopede of Eudoxus,the fact being confirmed by theapplication of the sameterm hippopede (horse-fetter) by Proclus to a planecurve of similarshape formedby a planesection of an anchor-ringor toreinternally and parallelto its axis.13 We knowthat Eudoxus, like Archytas his teacher,studied the intersections of spheres, cylinders,and cones,and thereis no reasonto doubtthe possibility that Eudoxus rec- ognizedSchiaparelli's spherical lemniscate. At the same time,Heath may have been too hastyin givingthis lemniscate such an exclusivehold on whatEudoxus might have called "hippopede."Note that what Proclus called a hippopedeis differentfrom Schia- parelli'sspherical hippopede. The spiricsection Proclus called by thisname is a plane figure.It cannotbe createdby theplane projection of Schiaparelli'shippopede, and it cannotbe transformedinto Schiaparelli's hippopede by projectingit froma planeonto a sphere.Proclus's hippopede, then, refers to a familyof curves that is distinctfrom the familyof curves referred to by Simpliciusas thehippopede of Eudoxus. The inevitable conclusionfrom this must be thatby way of a ratherloose, impressionistic analogy to an equestrianleg-cuff, the word "hippopede" describes a largevariety of self-intersecting curves,that may lie on plane,spherical, and perhapsother surfaces. As such,the curve shownin Fig. 11 belowis a perfectlylegitimate hippopede. This hippopede, and others likeit, are createdby thealternative interpretation in the special case whenthe inclinationbetween the axes ofspheres 3 and4 is equal to thelatitudinal position of the planet 14 measuredfrom the pole ofsphere 4. Theydiffer from the hippopedes of Schiaparelli's reconstructionin that they cannot be createdby the intersection of a spherewith a cylinder.The twofamilies of hippopedesare also distinguishedby thetangent at thepoint of selfcontact, which exactly coincides with the long axis of symmetryin thecase of thealternative hippopedes, while in thecase ofSchiaparelli's hippopedes this never occurs.A furtherdistinction resides in themotion of the point that traces these hippopedes as thegenerating spheres revolve: it crossesover to theother side of thehippopede's

12 ¡bid.,pp. 96-97(1 19.9-1 19.17). 13 Heath,Aristarchus of Samos, p. 207. 14 This willbe recognizedas thespherical case ofthe Tusi couple.As F.J.Ragep noted,there seemsto be no reasonto doubtthat al-Tusi was bothsupported and inspiredby Aristotle'stext in thegeneral venture of considering homocentric alternatives to certainPtolemaic arrangements, butit does notnecessarily follow from this that al-Tusi constructed his own modelas a specific interpretationof Aristotle's passage (F.J.Ragep, Nassir al-Din al-Tusi's Memoiron Astronomy (al-Tadhkiraficilm al-hay 'a), 2 vols.,(New York:Springer- Verlag, 1 993), p. 33). Indeed,al-Tusi's failureto noticethat in thespherical case theTusi coupletraces a self-intersectingloop and not merelylinear oscillatory motion along a greatarc suggeststhat he conceivedof the spherical versionas an extensionof the co-planar case wherethe resulting motion is an oscillationalong a straightline without any latitudinal deviations.

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Figure1 1

Figure12

Figure1 1 showsthe curve created by the alternative interpretation, with sphere 4 inclinedby 17.5° tosphere 3, withthe planet on latitude1 7.5° measuredfrom the pole ofsphere 4, andwith a phase angleof 5°. Figure12 showsthe trace of theresulting planetary motion with an eclipticperiod threetimes longer than the synodic period

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions 236 I. Yavetc longaxis of symmetryat thepoint of self-intersectionin Schiaparelli's case (Fig. 13a); in thealternative case, itswings back from the intersection point, and crosses the axis of symmetryonly at thetwo extreme turning points of the loop (Fig. 13b).That both types of hippopedesare equallylikely to have been consideredby Eudoxusappears to be as plausiblea conclusionas anyto drawfrom the testimony of Simplicius.15 To recapitulate,we haveseen that with the word hippopede, we mayuse Simplicius's testimonyto restrictthe alternative interpretation of Aristotle's text to thespecific case wherethe two inclinations are always equal. We havealso seenthat this does notsuffice to rejectthe alternativeinterpretation in favorof Schiaparelli's.Indeed, Simplicius's textcannot be read as a geometricalblueprint containing a set of instructionsfor the constructionof Schiaparelli'sarrangement to theexclusion of anyalternative. Quite to thecontrary, Simplicius's text is ratherambiguous at severalcrucial points, and it is only a reverse-interpretationguided by a preconceivednotion of hippopede as theintersection ofa sphereand a cylinderthat leads to our understanding ofthis text in the exclusive terms of Schiaparelli'sreconstruction. It then takes another 900 yearbackward implication to imposethis restricted interpretation on Aristotle'stestimony. The laststep is justifiable onlyon theassumption that Simplicius's description of theEudoxian system is more authoritativethan Aristotle's. It is truethat Simplicius supplied more information than

0V Viii A A 0 A a b

Fig. 13. The motionof the tracing point in Schiaparelli'sreconstruction (a) andin thealternative reconstruction(b)

15 O. Neugebauerreportedly pointed out that the figure of 8 tracedby the spherical Tusi couple cannotbe considereda hippopedebecause the tangent at thepoint of self-intersectionis 0° (F.J. Ragep, Nassir al-Din al-Tusi's Memoiron Astronomy(al-Tadhkira fi dim al-hay'a), 2 vols., (New York:Springer- Verlag, 1993), p. 455 (note)).This, however, presupposes the exclusivity of Schiaparelli'sreconstruction without independent justification on the basis of the historical record.

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Aristotle,but as we shall see in thenext section, it does notfollow from this that his informationis necessarily more dependable than Aristotle's.

d. How Centralis theRole ofthe Hippopede in Eudoxus's HomocentricAstronomy?

It is remarkablethat the hippopede, which in thehands of Simpliciusand Schiapar- ellihas become the hallmark of Eudoxus's planetary theory, was notmentioned at all by Aristotle.This could have several explanations, all ofthem both plausible and tentative owingto thegeneral lack of sourcematerial that typifies the historical analysis of early Greekthought. Aristotle's own contribution to theEudoxian theory was in addingcon- nectingspheres between the four-fold sets of spheresfor consecutive planets, so as to turnthe entire scheme into a single,mechanically continuous system. For his purpose it sufficedto notethat the motion of each planetinvolved separate rotational compo- nentsthat had to be counteractedby theintervening spheres before the next planetary motioncould begin.16We mayadd to thisthe suggestion that while Aristotle ranked mathematicshighly as a subjectfor cultivation, mathematical detail does notfeature prominentlyin his work.Therefore, we shouldnot marvel to findthat he avoidedthe detailsof Eudoxus'sinherently mathematical construction. This depictionof Aristo- tle as notactively pursuing the details of mathematicalastronomy and merelytaking overwithout modification (albeit with some expressed doubts) the improved Eudoxian 17 modelof Callippus is quitecommon, and need not be furtherdeveloped here. It maybe

16 It has oftenbeen noted that Aristotle used superfluousspheres in hisconstruction by using a diurnalouter sphere for each individualplanet, despite the fact that this motion is automatically transferredfrom the previous planet whose distinctive rotational components have been canceled bythree counteracting spheres. To thebest of my knowledge, however, it has notbeen noted that froma purelygeometrical standpoint, Aristotle actually used two superfluousspheres for each planet.He coulduse twospheres to cancel outthe rotations of thehippopede-forming spheres 3 and4. The nextcounteracting sphere, with axis alignedwith the axis ofthe ecliptic sphere, should thenbe madeto rotatesuch that the sum of itsrotation and thatof theprevious planet's ecliptic spherewould amount to theecliptic period of thenext planet. Only two extra spheres are then neededto producethe hippopede (or alternative)of thenext planet. In short,Saturn would have fourspheres, then Jupiter would follow with three counteracting spheres plus two more spheres, as opposedto Aristotle's three plus four. Following this for the rest of the planets, the sun, and the moonwould yield a totalof 31 spheresas opposedto Aristotle's43. Speculationson thereasons forthe presence of unnecessary spheres in Aristotle'sscheme are outside the scope ofthe present discussion. 17 See e.g. MichaelJ. Crowe, Theories of the World from Antiquity to theCopernican Revolu- tion,(New York: Dover Publications, Inc., 1 990), pp. 26-27. J.L.E.Dreyer, A Historyof Astronomy fromThaïes to Kepler, 2nd edition, (Dover Publications, Inc., 1953; originallypublished in 1906), pp. 112-114.Thomas S. Kuhn,The CopernicanRevolution, (Cambridge, Mass.: HarvardUni- versityPress, 1957), p. 80. David C. Lindberg,The Beginningsof WesternScience, (Chicago: The Universityof ChicagoPress, 1992), pp. 95-96. G.E.R. Lloyd,Early Greek Science: Thaïes toAristotle, (New York:W.W. Norton & Company,1970), pp. 92-94. J.D North,The Fontana Historyof Astronomy and Cosmology,(London: The FontanaPress, 1994), pp. 82-84. S. Toulmin

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submitted,therefore, that Aristotle showed only marginal interest in the detailed descriptionof planetarymotion and thathe studiedplanetary astronomy primarily to evaluate itsplace in largercosmological and theologicalcontexts. These contexts,however, require none of thedetails Aristotle supplied regarding theplanes of revolutionof thevarious spheres, and shouldhave led to a non-specific descriptionthat could accommodate the homocentric schemes of Al-Bitruji, Amico, and Fracastoroin additionto Schiaparelli'sreconstruction and the alternatives suggested in theprevious sections.18 Each ofthese homocentric schemes could provide the basis for thetheological exercise of counting the number of overseeing deities, and each could be unifiedinto a grandmechanical scheme by meansof interconnectingcounter-spheres. Al-Bitruji'sscheme, however, has rotations relative to the ecliptic sphere about poles that are farremoved from the ecliptic, which explicitly violates Aristotle's description. The planetaryschemes of bothAmico and Fracastororequire more than four spheres with mutualinclinations that differ from those described by Aristotle, and this too violates his accountof the Eudoxian and Callippic systems. It appears,then, that Aristotle's descriptionof theEudoxian scheme outlines a restrictedclass of homocentricarrangements witha degreeof geometrical specificity that exceeds the requirements of his theological and cosmologicalmotivations. His neglectto mentionthe hippopede, then, cannot be writtenoff as stemmingfrom a generallack of interestin thespecifics of theEudoxian scheme.So thequestion arises again, why did Aristotlefail to mentionthe hippopede? Afterall, if Simplicius'sovert emphasis on thehippopede is justified,then Aristotle's accountappears like a descriptionof Kepler'ssystem without mentioning ellipses. For all we know,Aristotle may be guiltyof suchan oversight,but there also exists anotherpossibility. It is generallyacknowledged that Eudoxus's scheme represents the firstsophisticated Greek attempt to reduceplanetary motion to circularcomponents. It is also generallyadmitted that this attempt is furtherconstrained by the desire to achieve thereduction while maintaining perfect spherical symmetry and uniformityof motion in theheavens.19 Both Aristotle'sand Simplicius'sdescriptions of Eudoxus's system clearlypreserve these two fundamental requirements, but Simplicius's account is more restrictivethan Aristotle's - perhaps too much so. Itmay be thecase thatAristotle did not mentionthe hippopede because it represented only a limitedclass ofEudoxian systems, andbecause for his cosmological discussion Aristotle considered it important to outline theirshared principles rather than their particular differences. He thereforewrote an andJ. Goodfield, The Fabric of the Heavens: The Development of Astronomy and Dynamics,(New York:Harper & Row,Publishers, 1965), pp. 106-107. 18 Al-Bitruji,On thePrinciples of Astronomy, Translated and Analyzedby BernardR. Gold- stein,(New Haven: Yale UniversityPress, 1971). Noel Swerdlow,"Aristotelian Planetary As- tronomyin theRenaissance: Giovanni Battista Amico's HomocentricSpheres," Journal for the Historyof Astronomy, 3 (1972): 36-48. Mario di Bono, "Copernicus,Amico, Fracastoro and Tusi's Device: Observationson theUse and Transmissionof a Model,"Journal for the History of Astronomy,26 (1995): 133-154. iy Thiswas notalways the understanding of Aristotles passage.In 1795,for example, Adam Smithunderstood the third sphere in the Eudoxiansystem to oscillateback and forthlike the wheelpendulum in a clock (Adam Smith,Essays on PhilosophicalSubjects, Edited by W.P.D. Wightmanand J.C. Bryce, (Oxford: Clarendon Press, 1980), p. 58).

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions On theHomocentric Spheres of Eudoxus 239 account thataccommodated the structureof Schiaparelli's reconstructionas well as the alternativedescribed above because both of themhad been consideredby Eudoxus. The generalityof this account reflectsnot a basic lack of interestin astronomicalgeometry, but a considered attemptto emphasize basic common features,which did not include the hippopede. But if that were the case, we must ask ourselves why Simplicius did give the hippopede such a prominentplace in the Eudoxian scheme. A possible answer to this question lies in the assessment of Simplicius's reliability,which will be further examined here. We have already seen that Simplicius's seemingly more specific account actually suffersfrom several ambiguitiesof which Aristotle'stestimony is free.To these mustbe added Simplicius's problematicaccount of the Eudoxian lunar theory,which accounts for the motion of the moon by the combined rotationsof threehomocentric spheres. Accordingto Simplicius,the firstsphere rotates about the celestial pole every24 hours, thesecond rotatesabout theecliptic axis once everymonth, and thethird sphere, which is inclinedto thesecond, carriesthe moon and rotatesonce every18.5 years.Unfortunately, thiswould make the moon stay to one side of the ecliptic forabout nine years beforeit crosses to the otherside where it remains foran equal lengthof time,while in factthe moon crosses theecliptic twice a month.Martin in the 19thcentury, and Dicks in the20th, have taken Simplicius at face value as representingthe state of Eudoxus's knowledge of the moon's motion.20The error,however, is easily correctedby simply switching therotation periods of the second and thirdspheres withoutchanging the orientationof theirrespective poles. The majorityof historianstend to consider Simplicius's account as an incorrectrepresentation of the Eudoxian theory,and thisis also the point of view adopted here.21 Turningnow to Aristotle'saccount, we findit therestated that for the fiveplanets as forthe sun and the moon, the firsttwo spheres are the same: the firstspins in the plane of thecelestial equator,the second spins in theplane of the ecliptic. Keeping in mindthe observationsof theprevious paragraph, we notethat while thesecond sphererotates with theperiod of mean eclipticrevolution in thecase of thefive planets, it should notdo so in the case of the moon if properagreement with observations is desired. Some historians thereforesuggested that Simplicius's errorgoes all the way back to Aristotle,22but as we shall presentlysee, thiscan be justifiedonly by reading Simplicius into Aristotle- a ratherdubious procedurein this case. It has already been pointed out thatnowhere did Aristotlesay anythingabout the speeds of these spheres, so thatas far as it goes, his

20 D.R. Dicks, Early GreekAstronomy to Aristotle,(London: Thames and Hudson,1970), pp. 180-181,256 (note338). 21 It is unlikelythat Eudoxus could precisely position the ecliptic - beingthe trace of the sun's pathon thebackground of thefixed starts - and tracethe moon's path relative to it. The moon, however,executes significant and easily observable latitudinal oscillations relative to the zodiacal beltwithin the course of a singlelunation, resulting in an eclipsepattern that is inconsistentwith Simplicius'sdepiction. No referenceto theecliptic is requiredfor making this observation, and it does notseem very likely that Eudoxus was actuallythat ignorant of knowledgethat was within easyreach in hisday. Heath,Aristarchus of Samos, p. 197. Dicks (Ibid.),who took the error to be Eudoxus's,also readAristotle's passage as identicalto Simplicius's.

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descriptionis quiteunproblematic: in all cases,the first sphere rotates about the celestial axis,and thesecond sphere rotates about the ecliptic pole. Thisremains true of the first andsecond lunar spheres regardless of the speeds we mightwish to assignto thesecond and thirdspheres. Simplicius specified beyond Aristotle that in all cases the second sphererotates with the mean ecliptic period of themember it pertainsto, be it thesun, moon,or any of the planets. In thecase ofthe moon this amounts to a grossincongruence withthe observed appearances. But the only way to suggestthat this was also Aristotle's view,is by readingSimplicius's specification of rotationalspeeds into Aristotle's text. Thisis a possible,but by no meansthe only plausible, reading of these texts. It is equally plausibleto takethese problematic passages to suggestthat Simplicius's testimony is less trustworthythan Aristotle's. Knorrrecently observed that "the mere survival of a documentneed not compel us to believewhat it says."23This cannotbe ignoredconsidering the existence of several problematicinstances in Simplicius'scommentary on Aristotle'sdescription of the Eu- doxianhomocentric theory. In particular,it wouldnot be an isolatedmisrepresentation on Simplicius'spart to inappropriatelyemphasize the hippopede which Aristotle did noteven see fitto mention.Furthermore, in this particular case specificconsiderations regardingthe mathematical nature of the hippopede might have encouraged Simplicius to do so. The hippopedeof Schiaparelli'sarrangement possesses a greatmathematical valuebecause it can be generatedby the intersection of two simple solids - a sphereand a cylinder- and its entiretrace may thereforebe reducedto theclassical geometrical procedureof syntheticconstruction.24 This maybe thereason for the emphasis given to itby Simplicius.There seems to be no reasonto doubtthat Eudoxus constructed this curve,and used it as a rigorouslyanalyzable example of oscillatorymotion generated bythe combination of uniform homocentric rotations. It is notat all clear,however, that in Eudoxus'soverall astronomical work, the hippopede occupied a centralposition. Fromthe mathematical point of view, the curves generated by the alternative arrange- mentare notas attractiveas thehippopede. They may, of course,be generatedby the intersectionof a spherewith other surfaces. However, unlike the cylinder that generates thehippopede, the new generating surfaces cannot be definedindependent of the curve theycreate while intersecting a sphere. In otherwords, we cannotavoid treating the de- siredcurves as generatedby mechanicalmotion. In termsof rigorousGreek geometry, therefore,the alternative curves may have been considered problematic as mathematical entities,and progressivelymore so in thepost-Euclidean Hellenistic tradition. Indeed, Simpliciusseems to haverated mechanical construction as an inferiorapproach to ge-

23 WilburRichard Knorr, The Ancient Tradition of GeometricProblems, (New York:Dover Publications,Inc., 1993), p. 10. "Manyscholars doubt Simplicius's authority and question Plato's putativerole in thedevelopment of astronomicaltheory." B.R. Goldsteinand A.C. Bowen,"A New Viewof Greek Astronomy", ISIS, 74 (1983): 330-340, p. 330. 24 FollowingSchiaparelli, Heath showedhow in additionto a cylindricalintersection, the hippopedeis theintersection of a spherewith two cones joined at the apex, but this does not exhaust themathematical beauty of this curve. John North ("The Hippopede,"in A. vonGotstedter (ed.), Ad Radioes,(Stuttgart: Franz Steiner Verlag, 1 995), pp. 143-1 54) recentlyshowed that the intersection of a spherewith a parabolicsurface also producesthe hippopede of Schiaparelli'sinterpretation. This can easilybe demonstratedin modernanalytical terms as shownin theappendix.

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions On theHomocentric Spheres of Eudoxus 24 1 ometricalproblems.25 As an astronomicalentity, however, the alternativecurve is both legitimatein Greek usage and highlyuseful. We possess some evidence to the effect thatin the studyof geometricalproblems the Greeks were not exclusivelydependent on rigorousgeometrical analysis as exemplifiedin Schiaparelli's classical paper, but that theymade considerable use of physical models where rigorousanalysis failed them.In a well-knownparagraph, Plutarch specifically mentioned Eudoxus and his Pythagorean mentorArchytas in thisconnection:

For the artof mechanics,now so celebratedand admired,was firstoriginated by Eudoxus and Archytas,who embellished geometrywith its subtleties,and gave to problems incapable of proof by word and diagram, a supportderived from mechanical illustrationsthat were patentto the senses.26

Withelementary trigonometry, equally elementaryuse of thegroup of sphericalrota- tionsand a low-powerhome computer,it is possible to trydifferent setups of homocentric spheresand to generatevarious hypotheticalplanetary paths. Eudoxus, and othersthat workedwith his planetaryscheme, had neithersystematic trigonometric tables to execute the necessarycalculations nor personal computers.However, the sphericalarrangement describedin Fig. 4 may verywell have functionedas a crude analogue computer.It must be emphasizedthat there is no need to actuallybuild nestedhomocentric spheres for this purpose. A set of concentricrings, consecutively attached to each otherat diametrically opposed swivel points would suffice,and an object attachedto the innermostring can be made to demonstratethe basic motionof a planet accordingto any of the alternatives discussed thus far,including Schiaparelli's. A finerstudy of the traces generated by such motionsrequires no more than a single sphere and a compass fordrawing a set of pointson the spherethat may be joined to formcontinuous paths. It is time-consuming work,but not any more so than Kepler's manual calculations of planetaryorbits. We have no evidence regardingthe way Eudoxus and his immediatefollowers went about the practical business of studyingthe details of theirnewly inventedcinematic theoreticalastronomy. Consequently, we cannot reject out of hand the possibilitythat their theoreticalastronomy was more an analogue graphical than a rigorousgeometrical af- fair,whatever "rigorous" may have meantin those days. We do possess indicationsthat portrayEudoxus as interestedin the mapping of stars on solid spheres. Cicero, when recountingthe sack of Syracuse, described some booty broughtto Rome by Marcellus, thatcontained a spherical mechanizationof planetarymotion created by Archimedes. The mechanization,Cicero explained, was Archimedes's innovation,but plottingstars on solid sphereswas already an old tradition;and Cicero explicitlymentioned Eudoxus of Cnidus in thisrespect. It was, Cicero reported"... a veryearly invention, the firstone

25 Knorrargued at length(The Ancient Tradition of GeometricProblems, [New York:Dover Publications,Inc., 1993]), that this gradual shift toward formalization and rigorization is responsi- ble foran over-rigorizedimage of early Greek geometrical practices created by the doxographers ofthe early Christian era. 26 Plutarch'sLives, "Marcellus," XIV. 5-6. Plutarch'sstory that Plato stronglydisapproved of Eudoxusand Archytasfor using such mechanical devices, and ". . . inveighedagainst them as corruptersand destroyers of the pure excellence of geometry" is opento doubts.

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions 242 I. Yavetz ofthatkind having been constructed by Thaïes of Miletus, and later marked by Eudoxus of Cnidus. . . withthe constellations and starswhich are fixedin thesky . . . ,"27Of far greaterimportance are thepoem by Aratusthat supposedly describes Eudoxus's stellar sphereand thedetailed criticism of thispoem by Hipparchus,who claimed that Aratus closelyfollowed Eudoxus. It has beensuggested that the sculpted depiction of the stellar spherecarried by Atlas was also modeledon Aratus'spoem.28 Now, if Eudoxus devoted timeto mapping the fixed stars, the equator, the ecliptic, the tropics, and the Arctic circles on a sphere,it would not be outof character for him to try to plotat leastthe outstanding aspectsof planetarymotion, namely, the retrograde phases, on a sphere.Thus, in an astronomicaltradition that makes use ofboth physical models and geometrical analysis, inabilityto achieverigorous geometrical demonstration would not represent an insur- mountableobstacle to further progress. Analogue graphical "computations" in the above sense,using the alternative loop as thegenerator of retrogrademotion, would generate tracesthat do agreequalitatively with the observed paths traced by theplanets in their motion.If true,this suggests that we viewthe Greek astronomer in the4th century BC as a practicalgeometer interested more in thegeneration of astronomically useful forms thanin theirrigorous justification. Of course,without conflicting with this view of Greekastronomy, the successful syntheticdemonstration ofone modelof homocentric spheres, namely, the one thatcre- ates thehippopede of Schiaparelli'sinterpretation, could verywell encouragefurther studyand developmentof othersuch models.At the same time,it wouldbe natural fora doxographerof Simplicius'staste and inclinationsto givespecial emphasis to the mathematicallyjustified model as representinga first triumph in theattempt to bring sphericallysymmetric astronomy and rigorous geometry into harmony. About nine hun-

27 Cicero,De republica, I, XIV, 21-22. Dicks notedthat the attribution of sucha sphereto Thaïesdoes notadd credence to Cicero's account. It may also be mentionedthat Cicero reported on theArchimedean spheres in the course of recounting a conversation that may or may not have taken place in realitybetween Scipio Aemilianusand somefriends. The objectof the conversation was to presenta typicallyRoman emphasis on theoverriding importance of practical issues pertaining to publiclife, as opposedto Greekidealistic and abstractthought. When the older and highly respectedLaelius joined theconversation, he askedPhilus, who initiatedit: "Do youreally think then,Philus, that we havealready acquired a perfectknowledge of those matters that relate to our ownhomes and to theState since we are now seekingto learnwhat is goingon in theheavens?" and he continuedto statehis opinion that the usefulness of the abstract and mathematical sciences is that". . . iffor anything at all, they are valuable only to sharpen somewhat and, we maysay stir up thefaculties of the young, so thatthey find it easier to learn things of greater importance." After this, theconversation quickly came downto earth,but not before Laelius repeatedthe point attributed by Scipio to Socratesabout the unusefulness of naturalphilosophy. Under these circumstances, it is notsurprising to findmodern scholars expressing doubts regarding the reliability of Cicero's report.On theother hand, why lie aboutthe use ofsuch spheres? Indeed, why bring them up inthe firstplace? At thevery least, the passage reflectsthe existence of suchdevices in Cicero'stime, whiletheir attribution to Thaiesin theabove manner suggests some confusion regarding Thales's stateof knowledge.But regardingEudoxus, the poem of Aratus,its critique by Hipparchus,and thestatements by Plutarch suggest that Cicero used a realhistorical fact to makea politicalpoint. 28 J.B. Harleyand David Woodward,The Historyof Cartography,3 vols, (Chicago: The Universityof Chicago Press, 1987), vol. 1, pp. 140-143.

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dredyears separate Simplicius from the Aristotelian texts that he commentedon, and planetaryastronomy changed considerably in the interim. In Aristotle'stime the Eudox- ian homocentricspheres represented the state of theart in Greekplanetary theory. As suchthis theory was as importantas Eudoxus'scontributions tomathematics. Aristotle's text,which accommodates both the hippopede and its alternativeequally well, would fitnicely with this state of affairsif we actuallyallow it to includeboth interpretations. Indeedto Aristotle,who in theMetaphysics was demonstrablymore interested in the mechanizationthan in themathematization of astronomy,there would be no reasonto giveundue emphasis to one out of several known spherically symmetrical arrangements, merelyon accountof its mathematical neatness. By Simplicius'stime, planetary theory was dominatedby the Ptolemaicmodel, while Eudoxus's homocentricspheres were primarilyan ancientmuseum piece. Eudoxus'scontributions to geometry,on theother hand,were still as relevantas ever.If Simplicius'staste was morephilosophical than astronomical,he mayhave been naturally inclined to singleout the "mathematically correct"hippopede at theexpense of several other alternatives whose days of astronomical usefulnesshave long gone by.29

e. DifficultiesAssociated with the Astronomical Function of theHippopede

The reconstructionsdescribed in thisdiscussion assume that the primary purpose of thehippopede in Simplicius'saccount is to generatethe retrograde motion of a planet. To do that,the ecliptic component of themotion along the hippopede must exceed the rotationalspeed of thesecond sphere, which accounts for motion around the ecliptic. Now, the exteriorplanets Jupiter and Saturnretrogress many times during a single revolutionaround the heavens. This means that the third and fourth spheres rotate several timesfaster than the second sphere. In otherwords, the planet traverses the hippopede generatedby the third and fourth spheres several times during a singlerevolution of the secondsphere. Under such circumstances, motion even on a smallhippopede becomes swiftenough to produceretrograde motion, and, as Schiaparellishowed, hippopedes maybe definedfor these two planets to produce retrogressions ofthe correct arc lengths (see Fig. 14 and 15). The cases of Mars and Venus,by contrast,present severe difficulties because their ratesof revolution around the sun are comparable to thatof the earth. The case ofVenus is theworst. The synodicperiod of Venus is 584 days.Being an interiorplanet, its ecliptic motionmust be takeninto account by a spherethat rotates once in 365 days.This means thatits 3rd and 4thspheres rotate at 365/584, or 62.5% thespeed of its ecliptic (2nd) sphere. In otherwords, Venus completes nearly two revolutionsaround the ecliptic for every retrogradeepisode in itscomposite motion. In orderto givethe planet sufficient speed aroundthe hippopede so as to produceretrogressions, the size of thehippopede must

29 Thatphilosophical considerations favoring rigorous demonstrations over mechanical ar- rangementscolored the observations ofmany late Hellenistic commentators - Simplicius included - onGreek geometry isargued in W.R. Knorr, The Ancient Tradition ofGeometric Problems, (New York:Dover Publications, Inc., 1993), pp. 7, 364.

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Figure14

Figure15

Figure 14 shows the hippopedecreated by the equal and oppositerotations of two spheres aroundaxes thatare inclinedby 14° relativeto one another.When a planetmoving along this hippopedeis simultaneouslycarried around the ecliptic at a ratethat allows it to complete1 1 hippopedesin thecourse of one lap aroundthe ecliptic, it tracesthe course shown in Fig. 15. Jupiter'ssidereal period is 11 .86 years;its synodic period is 1.092 years,so itcompletes about 10.8 retrogressionsduring a singlecircuit of the ecliptic. Its deviation from the ecliptic in latitude is very small,and the arc lengthof its retrogressionsis on the orderof 15°. On the whole,then, the hippopedeappears to providea fairlygood qualitativerepresentation of itsmotion

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Figure16

Figure17

Figure16 showsa hippopedecreated by an inclinationof 87° betweenthe axes of the third and fourthspheres. Figure 17 showsthe planetary trace created when the planet completes 88% ofthe hippopede by the time its ecliptic sphere completes a wholerevolution, which represents the ratioof Mars's meanecliptic motion to its synodicperiod. Note how farthe planet is made to strayfrom the eclipticwhile it tracestwo retrogradeloops of small arc lengthin a single synodicperiod

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions 246 I. Yavety be increased (note thatthe planet will complete 5/8 of its hippopedal motionregardless of the hippopede's size duringone revolutionof the 2nd sphere). In particular,a large angle of inclination (at least 100°) between the axes of the thirdand fourthspheres mustbe specified.Unfortunately, this produces an undesirableincrease in themaximum angular breadthof thehippopede in additionto thedesired increase in its angularlength. Fortunately,Venus performsits retrogressionswhen it is close to the sun, which makes directobservation difficult. Martian retrogressions,by contrast,occur when Mars is in opposition to the sun, and are thereforenot similarlyrescued fromobservation. Mars circles the sun in 687 earth days and its mean synodic period is 780 days. That is to say, it must complete 88% of its hippopede by the time its ecliptic sphere completes a fullrevolution. The resultof attemptingto generateMartian retrogressions with a hippopede bears no resemblance at all to the observed ones. The hippopedal motion creates excessive deviations in latitude- dozens of degrees - fromthe ecliptic. Such deviations would place the planet well outside the zodiacal belt. Even the crudest of observationswould have disclosed it, but nothingof the sortever occurs (see Fig. 16 and 17), and Simplicius did note thatEudoxus was criticizedfor the motion in latitude generatedby his theory. This bringsto mindthe often-made statement that the hippopede's failurein thecases of Mars and Venus shows thatit served only as a qualitativeaccount of planetarymotion and as no more than a demonstrationof general feasibility.30There seems to be little reason to doubt thatEudoxus's homocentricastronomy served primarilyas a general, qualitative account and not as a precise quantitativemodel. At the same time,we ought to exercise some caution and be a bit more clear about what we mean by "qualitative." Hargreave's discussion suggests one meaning for"qualitative" in this connection.31To be even moderatelysuccessful, any planetary theorymust account for three readily observed phenomena: 1) All of the planets possess a mean eastward motionrelative to the fixedstars; 2) the eastwardmotion is periodicallyinterrupted by retrogradeepisodes during which a planet appears to move from east to west relative to the fixed stars; - 3) in addition to this motion in longitude - that is, along the ecliptic each planet shows some latitudinaldeviations fromthe ecliptic in the course of its motion. Look again at the theoreticalMartian path in Fig. 17 created by the hippopede in Fig. 16. It possesses a mean motion around the ecliptic, it shows distinctretrograde phases, and it certainlyproduces latitudinaldeviations fromthe ecliptic. So strictlyspeaking, there is no qualitativefailure in the above sense of the word in Eudoxus's theoryof Mars. At the same time,there is no denyingthe gross incompatibilityof thetheoretical and observable paths of Mars in retrogression.It appears, then,that Eudoxus's "qualitative" failurein the case of Mars boils down to these large deviations of theoryfrom observation.32 To

30 See, e.g. A. Aaboe,"Scientific Astronomy in Antiquity",Philosophical Transaction os the RoyalSociety of London, Series A, 276 (1974): 21-42, p. 40. 31 D. Hargreave,"Reconstructing the PlanetaryMotions of the EudoxeanSystem," Scripta Mathematica,28 (1967): 335. 32 Heathstated that Eudoxus's Martian theory fails to produceretrogressions because inclina- tionslarger than 90° are requiredfor the two inner spheres, and thatwould make them rotate in thesame direction. The problematicnature of this observation has beendiscussed in theprevious sections,but regardless of this discussion, Heath's observation (which is commonin the secondary

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions On theHomocentric Spheres of Eudoxus 247 rejectthe theoretical trace as qualitativelyunacceptable because of these deviationsis to compare the shape of thetheoretical trace to the observed traces witha definitecriterion for qualitative lack of congruence. Unfortunately,this poses the difficultproblem of deciding how large such deviationsmust be in orderto be called a qualitativefailure. Neugebauer appears to have used a somewhat differentsense of the word "qualitative"in this respect,which both sharpensthe issue and avoids the hopeless problem of settingquantitative criteria for qualitative differences.33Quantitative methods, he observed,appear to have become centralto Greek astronomyonly in the time of Hip- parchus,whose workwas significantlyenriched by exposure to Babylonian arithmetical astronomy.This suggestsa sense of the word "qualitative" as opposed to "quantitative" thatdoes notnecessarily equate "qualitative"with "imprecise." The concepts of geometrical congruenceand similarityare quite precise withoutbeing quantitative,and one may be qualitativelysensitive to verysubtle differencesbetween geometricforms. Subtle as theymay be, however,such differencesmust remain qualitative as long as no methodof quantifyingthe compared formsis available. This at once reveals the immense importance of Hipparchus'snew quantitativeastronomy, without rendering the old qualitative astronomyinsensitive to small differencesof geometricalform. The distinction,then, is betweenquantitative and qualitativemodeling, but "qualitative" in thissense does notmean "grosslyinaccurate." The hippopede certainlyproduces retrogrademotion, but for all intentsand purposes,it is still a qualitativefailure. Fig. 19 shows the retrogradetrace generatedby a hippopede that creates threeretrogressions per ecliptic revolution,which is about the ratio Eudoxus used for Mars according to Simplicius. Being 200% offthe mark, this ratio is questionable, consideringthat current values forthe synodic and ecliptic periods of all the otherplanets are well within10% of the Eudoxian values, as the table below shows. Both Ideler and E.F. Apelt, who an- ticipatedmuch of Schiaparelli's reconstruction,attributed the aberrantMartian synodic period to a scribe's error.34Schiaparelli was more cautious, and quite reasonably pre- ferredto suspendjudgement on the issue in the absence of otherevidence.35 As Heath

literature)is simplyfalse: as Figs. 16 and 17 above show,an inclinationof 87° clearlyproduces two smallretrograde phases in thepath of Mars. One maychoose to followSimplicius's lead, and nailEudoxus's failure to thewidely exaggerated deviations from the ecliptic, which exceed theobserved ones by about 25°. However,consider that Mars's retrogradearc can be up to 19.5° long,while the retrogression produced by the Eudoxian theory as depictedin Fig. 17 is onlyabout 2°-3° long.Eudoxus's theory of Mars is an orderof magnitude off both in longitudeand latitude. The largedeviations in latitudeare perhaps the easiest to singleout, but the traditional emphasis on latitudinaldeviation in thiscontext should be takenas a matterof conveniencemore than an identificationofthe fundamental problem. 33 O. Neugebauer,The Exact Sciences in Antiquity, 2nd edition, (New York:Harper & Brothers, 1957),pp. 156-162. 34 Ideler,"Uber Eudoxus (Zweite Abtheilung)," Abhandlungen der Königlichen Akademie der Wissenschaftenzu Berlin,(1830): 49-88, p. 78. E.F. Apelt,"Die Sphärentheoriedes Eudoxus undAristoteles," Abhandlungen der Friesischen Schule, Zweites Heft (Leipzig: Verlag von Wilh. Engelmann,1849): 27-49, p. 42. 35 G.V. Schiaparelli,"Le SfereOmocentriche di Eudosso,di Callippoe di Aristotele,"in Scritti Sulla Storiadella Astronomia Antica, 3 vols.,(Bologna: Nicola Zanchelli,Editore), voi. 2, p. 71.

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Planet Synodicperiods eclipticperiods Eudoxus modern %error Eudoxus modern %error Saturn 390 days 378 days 3 30 years 29 years166 days 2 Jupiter 390 days 399 days 2 12 years 11 years3 15 days 1 Mars 260 days 780 days 200 2 years 1 year322 days 6 Venus 110 days 116 days 5 1 year 1 year 0 Mercury 570 days 584 days 2 1 y ear 1 year 0 alreadynoted, a 3:1 ratioof synodicto eclipticperiod would make Mars undergotwo superfluousretrograde phases out of opposition in additionto theone in opposition,and thatis undoubtedlythe most significant discrepancy associated with this ratio.36 How- ever,even ignoring this discrepancy and restricting the comparison only to the theoretical and observedforms of the retrograde phase itself, the inadequacy of the hippopede im- mediatelystands out. Basically, this is becausea figureof 8 is notqualitatively similar to a simplestretched loop. In mostcases, a retrogressingMars describes a simpleelongated and slightlyskewed loop on thebackground of the fixed stars. The entireloop is located to one side of theecliptic. By contrast,retrograde motion generated by Schiaparelli's hippopedeis alwaysexactly symmetrical with respect to theecliptic, which is crossed by thetheoretical planet precisely in themiddle of theretrograde track. Furthermore, theretrograde path itself usually looks likea doubleloop thatbears no resemblanceto theobserved motion.37 It takesno morethan a scoreof randomobservations of Mars duringthe 5 monthsit takesto traceits retrogradeloop to see thata hippopedecould neverproduce an acceptablesemblance of its trace, even if the hippopedal path has the correctarc length(compare Figs. 18 and 19). The onlytime that a hippopedecould createa veryrough semblance of a planet's actualretrograde path is whenthe middle of theretrograde phase occurs near the intersectionof theplanet's path with the ecliptic, in whichcase theretrogression ceases to looklike a loop,and assumes the form of a simplezig-zag motion. On theaverage, Mars tracessuch paths in about3 outof 10 retrogressions.A closer comparison of such a path withthe one created by a hippopede(see Figs.20 and21) againshows that because of the hippopede'scrossing self-intersection and orientationrelative to theecliptic component ofthe motion, it makes the planet curve into and outof the retrograde track in a manner thatcannot be reconciledwith the observed trace. These qualitativeincompatibilities between the observed and theoreticalpaths remain blatanteven whenthe observedpath is tracedonly relative to the local group of starsthat form its immediatebackground without reference to theecliptic, which maynot have been mappedout with sufficient accuracy for critical comparison with a

36 Heath,Aristarchus of Samos, p. 210. 51 See O. Neugebauer,A Historyof AncientMathematical Astronomy, Part 111,(Berlin: Springer-Verlag, 1 975), pp. 1255-1 256 forillustrations of retrograde paths for Jupiter and Saturn. Mars's retrogradeloops areconsiderably wider and moreeasily observable.

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Fig. 18. Mar's betweenx^xLeo and Virgo,10/96-8/97; arc lengthM 9°

Fig. 19. Sectionof thestellar sphere (solid front,dotted back) showingparts of a hippopedal patharound the ecliptic, with a retrogradeloop (arc length^20°) createdby a 35° hippopede witha period3 timesshorter than the period of ecliptic revolution planetarypath in Eudoxus's time. In otherwords, it is difficultto see how the discrepancies between the observed and theoreticalforms can be swept under the rug by any sensible use of theterm "qualitative agreement" in thiscontext. In principle,and regardless of the particulardifficulties introduced by the ratios of synodic to ecliptic periods forMars, Venus, and Mercury,Schiaparelli's version of the hippopede simply cannot reproducethe basic formof planetaryretrogressions. Indeed, Schiaparelli himselfem- phasized precisely this qualitative incongruenceof formin his general discussion of

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Fig. 20. Marsin Libra,12/83-10/84 n7 „ - : i

Fig. 21. Sectionof the stellarsphere (solid front,dotted back), traversedby a 40° hippopede makingtwo retrogressions per ecliptic revolution theEudoxian theory,38 but he was notgenerally followed in thisby laterhistorians of science.Heath, to give but one example,devoted 3 pagesto quantitativecomparisons of observedand predictedarc lengths and breadths, without once notingthe formal incon-

38 G.V. Schiaparelli,"Le SfereOmocentriche di Eudosso,di Callippoe di Aristotele,'*inScritti Sulla Storiadella Astronomia Antica, 3 vols.,(Bologna: Nicola Zanchelli, Editore), voi. 2, pp. 75- 76.

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions On theHomocentric Spheres of Eudoxus 25 1 gruencebetween the theoretical and observedcurves.39 Now, Eudoxus was arguablythe greatestgeometer of his time,and much of his work involved the studyof geometrical forms- not merelystraight-edged ones, but curved formsfor the creation of which he constructedfairly elaborate instruments.To claim thathe never engaged in systematic quantitativeobservations of planetarymotion is one thing;it is quite anotherto claim that as a geometricianof his statureEudoxus was insensitiveto the incongruencebetween the trace generatedby a moving hippopede and a rudimentaryobservation of Mars's actual path. This does not add up to a sufficientreason forrejecting Schiaparelli's interpretation,but the significance of his interpretationmust be clear - it implies thatEudoxus neverbothered to take more thana casual look at Mars in retrogression. In the past decade, Bowen and Goldstein produced a series of studies on the nature of early Greek astronomicalobservations.40 For the purposes of the presentanalysis, therelevant conclusion fromBowen and Goldstein's studies is thatit would be unlikely forEudoxus or Callippus to map and date the fullcourse of Mars relativeto the ecliptic with the precision required for a critical comparison with the theoreticalalternatives described in this paper. That is to say, any discrepancyin a pointwise comparison of the theoreticaland observed position of the planet at a given date relative to some global mapping scheme would fall well withinthe range of observationalinaccuracy in Eudoxus's time. However, a differentpicture emerges if it is admittedthat rather than the precise reproductionof the planet's location in real time, the desired object was to reproduce the basic formof the planet's retrogradepath. To begin with, this harmonizeswith the assumptionthat the Eudoxian (as well as theCallippic) scheme was a qualitativerather than a quantitativeplanetary theory. As interpretedhere, the Eudoxian and Callippic theorieswere designed to addressthe departures of planetarymotions from uniformrotation. In thisframework, retrograde motion is clearlythe primary problematic phenomenon,not the predictionor retrodictionof a planet's coordinatesat a given date. Significantlyenough, observational procedures that fall considerably short of Bowen and Goldstein's standardsfor being consideredas precise, dated observationsare more than sufficientlyprecise to reveal the inadequacy of the hippopede as a realistic retrograde generator.In otherwords, observational practices in violation of Bowen and Goldstein's conclusions need not be attributedto Greek astronomersof the 4thcentury BC in order for them to have shown thatthe hippopede could not pass the test of observation.To see this, consider thatMars's retrogradeloops are not tinyforms that require refined instrumentsto be observed. They varyin arc lengthroughly from 1 Io to 19°, and their breadthranges up to 2.5°. Using threeor fourprominent stars in theconstellation against which Mars retrogresses,the formof its path can be easily traced by a small numberof observationsunsystematically spread over the course of fivemonths without ever needing to measure angles in excess of 10°. No particularstandard of angular measurementis requiredfor this, nor does the planet's position need to be fixed relativeto any global

39 Heath,Aristarchus of Santos, pp. 208-210. 4U Of particularimportance to thisstudy are "Hipparchus'Treatment of EarlyGreek Astron- omy:The Case of Eudoxusand theLength of Daytime,"Proceedings of theAmerican Philo- sophicalSociety, 135 (1991): 233-254,and "The Introductionof Dated Observationsand Precise Measurementin GreekAstronomy" Archive for History of Exact Sciences, 43 (1991): 94-132.

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coordinatesystem or a timedtable of stellarrisings and settings.Because of thesmall anglesinvolved, the difficulties associated with the sectioning of a circledisappear, and theproblem reduces to one ofmeasuring linear separations on a practicallyflat patch of sky.In fact,a T-bar1 meterlong, with a 15 cm longcross beam on which16 equidistant segmentsare marked provides a primitivedioptra capable of measuring small angles by incrementsof roughly 0.53°. Timemeasurements of anykind are not required. In other words,significantly less sophisticatedobservational techniques of a sortthat is notruled out by theconclusions of Bowen and Goldsteinsuffice for clearly demonstrating the inadequacyof Schiaparelli'sarrangement as a basis foraccounting for the appearances. Weshall see thata similarcomparison would also indicate some deviations from observed resultson thepart of thealternative reconstruction of Eudoxus'stheory. However, as a steptoward a qualitativetheory the alternative certainly appears to pointin theright direction,and, we shallalso see, itcould provide Callippus with a startingpoint to a far bettertheory than the one ascribedto himby Schiaparelli. All ofthis does notnecessarily mean that the hippopede was completelyunimportant in Eudoxus's astronomy,or thatSimplicius's text is totallymisleading, or thatSchia- parelli'sreconstruction should be rejected.It does notnecessarily follow that Eudoxus and his contemporariescarried out certain observations just because suchobservations werewithin their capabilities. It does suggest,however, that if Schiaparelli's hippopede was centralto the Eudoxian astronomical system, then Eudoxus's interest in elementary observationalastronomy was at bestmarginal. It appearsmore likely in thiscase, that forEudoxus the problem of accounting for planetary motion was no morethan a spring- boardto a purelymathematical problem of analyzing superimposed spherical rotations. We have seen thatregardless of its astronomicalusefulness the hippopede represents a triumphof geometricalanalysis. Furthermore, Riddell has shownthat Schiaparelli's arrangementwith the third sphere rotating twice as fastas thefourth may be used as a mechanicalcontrivance for solving the problem of doublingthe cube.41 This maybe used to portraya Eudoxuswho transported a primarily mathematical device into an as- tronomicalcontext as a quickafterthought that he hadno particularinterest to followup withobservational comparisons. Unfortunately, we do notreally know what Eudoxus's attitudeto observationalastronomy was. Straboreported that Eudoxus had observato- ries in his homeisland of Cnidusand in Heliopolis,Egypt, which still existed in the timeof Augustuswhen Strabo wrote about them.42 If thepoem of Aratusis based on

41 R.C. Riddell,"Eudoxan Mathematicsand the Eudoxan Spheres"Archive for Historyof ExactSciences;9 20 (1979): l-19,esp.pp. 14-17. 42 TheGeography of Strabo, 8 vols.,Tr. H.L. Jones,(London: William Heinemann, 1908), vol. VIII, pp. 83-85 (C 806-C 807), ". . . fora kindof watchtower is to be seen in frontof Heliupolis, as also in frontof Cnidus,with reference to whichEudoxus would note down his observations of certainmovements of theheavenly bodies." Strabo's story that Eudoxus visitedEgypt with Platofor thirteen years, no less (alternativereadings of three years and sixteenmonths have been suggested),seems rather questionable. But Strabo's indication that Eudoxus occupied himself with astronomicalobservations of some sort need not be discardedalong with the less credibleparts of thestory. A. Berry,A ShortHistory of Astronomy, (New York:Dover Publications, 1961, reissue of theoriginal 1898 text),p. 29, tookStrabo's testimony at face value,and thoughtof Eudoxus as an activeobservational astronomer. Paul Tannery,Recherches Sur L'Histoire De L'Astronomie

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observationscarried out by Eudoxus,then he did have morethan a passinginterest in activeastronomical observations. We knowthat Eudoxus devoted a treatiseto thetheory of homocentricspheres, but neitherit, nor any otherastronomical work by Eudoxus and Callippushas survived. Latersources, particularly Ptolemy, do notrefer to anyobservations by Eudoxus and his contemporariesfor the purpose of ascertainingthe form of planetaryretrograde loops. Butthen, the sort of observationsthat Eudoxus was likelyto carryout for this purpose wouldhave been fartoo crudeand imprecisefor Ptolemy's more refined purposes, so he hadlittle motivation for citing such old knowledge,even if it did exist.In theabsence of any extantreferences to observationscarried out by Eudoxus forthe purposeof ascertainingthe general shape of retrograde loops, we wouldnormally conclude that he didnot, in fact,engage in suchobservations. However, the general loss ofnearly all the astronomicalworks from his period requires that we qualifysuch statements as follows:It seemsmore likely that Eudoxus did not embark on an observationalprogram designed to elucidatethe form of planetary retrogressions, but technically they were certainly within hisreach, and for lack of sufficient evidence we cannotreject the possibility completely. Consideringthis, it seemsunjustifiable to close our historicalview to thepossibility thatobservational planetary astronomy was ofmore than marginal interest to Eudoxus, and thatconsequently the hippopede as reconstructedby Schiaparellicould notplay a centralrole in hisplanetary theory.

f. AstronomicalAdvantages of the Alternative Interpretation

Fromthe astronomical point of view, the alternative arrangement possesses several decidedadvantages over Schiaparelli's, and goes a considerableway toward alleviating thedifficulties outlined in theprevious section. This is particularlytrue of thegeneral alternativeinterpretation of Aristotle's text, but even the restricted alternative based on Simplicius'stext creates more realistic depictions of planetaryretrogressions than the hippopedeof Schiaparelli'sreconstruction. In the general case, theavailability of two independentinclinations instead of one makesit possible to createsimple long and nar- rowloops lyingalong theecliptic as opposedto theself-crossing figure of eightthat characterizesthe hippopede of Schiaparelli'sinterpretation. The additionof phasean- glesto thealternative arrangement skews the resulting loops relativeto theecliptic (see Figs. 8 and 9). Whencoupled to a rotatingecliptic sphere, this alternative arrangement givesrise to simpleelongated and slightlyskewed retrograde loops resemblingthe ob- servedones (see Fig. 10). We haveseen in theprevious section that the hippopede fails to providean adequatesemblance of Martianretrogressions even if thegrossly erro- neoussynodic period of 260 daysis taken.This is no longerthe case in theframework ofthe alternative arrangement. The loop it generatesstill diverges from the real one in

Ancienne,(Paris: Gauthier- Villars & Fils,1893, Reprinted by Arno Press, New York, 1976), pp.44-45, who was equally aware of Strabo's testimony, remained unconvinced, and considered thatEudoxus's many interests suggest that he did not possess the patient character required for lengthysystematic observations.

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions 254 I. Yavetz detailssuch as theparticular curvatures of its upper and lower sections, the point of self- intersection,and itsgeneral position relative to theecliptic (see Figs. 22 and 23). But it clearlypossesses the correct qualitative form and all of theabove discrepancies require farmore precise observations and pathanalysis to perceivethan those that characterize thecomparison with the hippopedal traces.43 Having said all ofthat,it is notvery likely thatEudoxus actuallyused 260 days forthe mean synodicperiod of Mars,when the real figureis 780 days.When this number is used,the alternative interpretation, either in itsrestricted or generalform, clearly diverges from observed Martian retrogressions. It is possible,for example, to enlargethe inclinations and to produceretrograde motion withoutunacceptable departures from the ecliptic. However, Figs. 24 and25 belowshow thatas theinclinations increase, they accentuate the differences between the geometry of a plane and thegeometry of a sphericalsurface. In excess of a certainsize, theal- ternativeloop looks like a rubberband that overlaps itself twice instead of themerely largerellipse that a deferentand an epicyclewould create on a planesurface. A proper choiceof phaseangle will produce a loopedretrogression, but the basic discrepancyof formclearly reveals itself by excessivemotions in latitudethat prevent the loop from assumingthe long flatshape of typicalplanetary retrogressions. Hence, while the alternativearrangement satisfies the general requirement of qualitativerepresentation of planetaryretrograde motion, it stillfalls short of reproducingthe particulars of Mars.

Fig. 22. Marsbetween Leo and Virgo,10/96 - 8/97;arc length-19°

43 By varyingthe parameters (e.g. planeton latitude17.5° measured from the pole ofsphere 4, inclinationof 18.5°between spheres 3 and4, anda phaseangle of 4°), thealternative arrangement can yielda farbetter qualitative semblance of thezig-zag path (shown in Figure20 above) than Schiaparelli'shippopede. The zig-zag pathcan also be producedby therestricted form of the alternativearrangement (e.g. 18° forthe inclination of spheres3 and 4 and forthe latitude of the planeton sphere4, and a phaseangle of 5°).

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Fig. 23. The retrogradepath generated by the alternative arrangement, with the planet carried on latitude17° (measuredyyfrom the pole) of sphere4, an inclination of 19° betweenthe axes of spheres3 and4, and a phaseangle of 2°. The ratioof ecliptic to synodicperiod is 730/260

From the point of view of historicalreconstruction, this is precisely the sort of effect we need, because historicalreconstructions of Eudoxus 's work that do allow him to account forthe motionsof the innerplanets would be hard put to explain the references to Callippus, who found it necessaryto amend Eudoxus's theoryfor Mars, Venus, and Mercury.Thus, contraryto the unavoidable implicationof Schiaparelli's interpretation, we now have a Eudoxus who did capturethe basic qualitativeform of planetary retrograde

Figure24

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Figure25 Figure24 showsthe curve that results from the combined motion of the alternative arrangement withthe planet on latitude40° as measuredfrom the pole of thefourth sphere, while the axes of the thirdand fourthspheres are inclinedby 49° to each otherand a phase angle of 4° is used.Figure 25 showsthe trace that a planetleaves on thestellar sphere if it completes 88% ofthe curvein Fig. 24 by thetime that its eclipticsphere completes one fullrevolution in accordance withMars's ratioof siderealto synodicperiod

Figure26

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Figure27

Figure26 showsthe curvegenerated by the alternativearrangement, restricted to two equal inclinations,44° each, witha phase angle of 4°. Figure27 showsthe resulting planetary path aroundthe ecliptic, when the Martian ratio of .88 forthe sidereal/synodic periods is chosen motion,but left it forCallippus to producea satisfactoryaccount for the actual motion ofthe three inner planets.

g. What Callippus May Have Done

We knowpractically nothing about the details of Callippus's contributions, save the generalstatement that he addedtwo sphereseach to themotion of thesun and moon, and one sphereto each of Mars,Venus, and Mercury.44For Mars, Schiaparelli showed howan additionalsphere can createa curvemore complicated than the hippopede with a constructionthat is a clevervariation on hisbasic arrangement.45The firsttwo spheres, as before,account for the diurnal rotation and formotion in theecliptic plane. Again

44 See Heath,Aristarchus of Samos, pp. 212-213. 45 G.V. Schiaparelli,"Le SfereOmocentriche di Eudosso,di Callippoe di Aristotele,"in Scritti Sulla Storiadella Astronomia Antica, 3 vols.,(Bologna: Nicola Zanchelli, Editore), voi. 2, pp. 79- 81.

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as before,the thirdsphere's poles are attachedto the second sphere at the ecliptic. The fourthsphere is inclined by 90° to the third,i.e., the pole of the fourthsphere rests on the equator of the third.The fifthsphere's pole is inclined at 45° relativeto the pole of the fourthsphere, in the directionof the pole of the thirdsphere. In otherwords, the startingposition has the pole of the fifthsphere halfwaybetween the pole of the third and fourthspheres. The planet restson the equator of the fifthsphere. The fourthsphere rotatestwice as fastas the third,in the opposite sense of the thirdsphere's rotation.The fifthrotates at the same speed and directionas the third.(Note thatby fixingthe pole of the fifthsphere directly under the pole of the fourth,the fifthsphere is made to rotateat the speed of the third,but in the opposite direction.In otherwords, this would create a hippopede). The resultingcurve lies symmetricallyalong the ecliptic. It is just over 90° long, and contains two triplepoints near its ends as Fig. 28 below shows. The advantage of this curve is twofold. To begin with,it is narrow,so that the planet's deviation in latitudedoes not exceed the observed deviation.Then, since theplanet is made to spend a long time near the ends of the curve because of the extraloops it must performthere, its motionalong the interiorpart of thecurve betweenthe two triplepoints is veryquick. Therefore,this relativelysmall curve can generate retrogrademotion even when the ecliptic period of the planet is shorterthan the synodic period (which is the period of timeit takes to complete a circuitof theretrograde generating curve). Retrogrademotion may be obtainedwith this new curve forMars, thathas a ratioof 0.88 betweenits ecliptic and synodic periods (see Figs. 28 and 29). - As Dreyer noted, the new curve crosses the ecliptic eight times unlike the hippopede, which crosses theecliptic fourtimes.46 In thecourse of a single synodicperiod, therefore,the planet would oscillate across the ecliptic several times more than it ac- tuallydoes. The observationalrejection of such a deviation,however, requires careful plottingof the planet's course around the entireecliptic and comparison of this plot withthe sun's motion.This involves systematicobservations that are considerablymore demandingthan the few random observationsrelative to a single zodiacal constellation thatsuffice to portraythe formof the retrogradephase itself.Unfortunately, even such a limitedobservation of just the retrogradeportion of the planet's course is incongruous with the theoreticalpath created by Schiaparelli's arrangement,because planets never curve in and out of theirretrograde phase as the model suggests. Furthermore,Mars in particulartraces distinctloops, which Schiaparelli's arrangementcannot reproduce.In the case of Callippus, thisrequires us to stretchthe meaning of "qualitative" agreement with the phenomena even beyond the level of discomfortassociated with the case of Eudoxus. Recall that Callippus allegedly amended Eudoxus's solar theorybecause it yielded seasons of equal length.Callippus added two spheres to the sun's motion,presumably to create a hippopede (or an alternativeretrograde generating loop) much too small to produce retrogrademotion, but sufficientto slow the sun down in part of its course and speed it up in the remainingpart so as to produce four seasons of unequal length.It does not seem reasonable thatthe same Callippus, who supposedly worried about disagreementsof one or two days out of 90 in the case of the sun, would allow

46 J.L.E.Dreyer, A Historyof Astronomy from Thaïes to Kepler, (New York: Dover Publications, Inc., 1953 reprintof the2nd edition, 1906), pp. 104-106.

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Figure28

Figure29

Figure28 shows the trace createdby the arrangementused by Schiaparellito reconstruct Callippus'svariation on theEudoxian theme. Figure 29 showsthe path for Mars generated by this arrangement.Such traces,as explainedabove, occur only when the planet crosses the ecliptic duringthe retrogradephase. Note how the modelmakes the planetcurve into and out of the retrogressionin a mannerthat is neverencountered in reality

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions 260 I.Yavety significantlymore blatant discrepancies between the theoretical and observedtraces of planetaryretrograde motion. It shouldbe notedthat the arrangement Schiaparelli used to reconstructCallippus's amendmentis usefulonly within a narrowrange: it producesa retrogressiongenerator in theform of a relativelylong (approximately 90° of arc),narrow closed curvealong the eclipticif the angle betweenthe fourthand fifthspheres is 45°.47 However,for inclinationsof 20 or 70 degrees,the arrangement creates quite unacceptable forms as Figs.30 and 31 show.In otherwords, while the 90°, -45°, 90°arrangement above could providebarely acceptable retrograde motion for Mars, its one variableparameter (being theangle between the poles of thefourth and fifthspheres, set hereto 45°) cannotbe variedto accommodatethe cases of Venusor Mercury. EchoingSchiaparelli, Heath referred to thisarrangement as thesimplest of several possiblealternatives which he didnot proceed to describe. Indeed, other variations using a fourthsphere that rotates twice as fastas thethird, and a fifthsphere that rotates at thesame speed and the same direction as thethird are made possible by abandoning the restrictionof theangle of inclinationbetween the third and fourthspheres to 90°. For example,instead of the90°, -45°, 90°arrangement that Schiaparelli showed, use 27°, 53°, and90°. Thisyields a hippopedethat cannot be reducedto intersecting spheres and cylinders.It is verynarrow, but spans an arcof 170° alongthe ecliptic. This would create a differentlyshaped retrogression for Mars (see Figs.32 and33). Thisform is acceptable foran ecliptic-crossingretrogression. As alreadynoted, however, such retrogressions arerelatively rare. In contrastwith these variationson Schiaparelli'sreconstruction, the alternative schemecreates none of the above difficultieswhen taken as Callippus'spoint of de- parture.The additionof a fifthsphere to thealternative interpretation is analogous to the developmentof the nextterm in a seriesexpansion, and its effectis intuitively explicable.The additionalsphere, placed beneaththe fourth sphere of theoriginal arrangement,makes the elongated, ellipse-like curve carry an additionalepicycle to which theplanet is attached.On theecliptic, the long axis of theoriginal curve is added to the(angular) radius of theepicycle, while at 90° fromthe ecliptic the short axis of the curveis subtractedfrom the epicycle's radius. This createsa longerand narrowerloop as shownbelow. With a smallphase angle as describedin sectionc above,the new curve maybe slightlyrotated relative to theecliptic. By specifyingthat 88% of thisloop be traversedwhile its center of symmetry completes one revolutionaround the ecliptic, we can produceretrograde motion in accordancewith the basic characteristicsof Mars's motion(see Figs.34 and35). The graphsat the end of the appendix show that every form ofMartian retrogression can be quitenicely reproduced with such a four-spherearrange- ment.The positionof theretrograde loops relativeto theecliptic, and thespeed of the

47 Schiaparellinoted that several alternative 3-sphere structures can createretrograde gener- atingcurves in keepingwith Callippus's needs, but did notproceed to describethem. Instead, he described one arrangementand characterizedit as the". . . simplestway which preserves the only " naturallimits of the [motionin] latitude.. . G. Schiaparelli,Scrìtti Sulla Storiadella Astrono- miaAntica, Vol. 2, (Bologna: Nicola Zanichelli,Editore), p. 79. BothDreyer and Heathsimply reproducedthe figures from Schiaparelli's original paper.

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Figure30

Figure31

Figure30 showsthe curvecreated by incliningthe fifthsphere by 20° relativeto the fourth in Schiaparelli's arrangement.The similaritybetween this curve and Schiaparelli's reconstructionofEudoxus's hippopede is deceptive:unlike the hippopede, this figure of 8 is created by intersectinga sphere with a pipe thathas an oval cross-section,not a circularone. Figure31 showsthe curveresulting from inclinations of 90°, -70°, and 90°. Both generateretrograde tracesthat are unacceptablefor reproducing planetary motions

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Figure32

^^^2^^ Figure33

Figure32 showsthe modifiedfigure of 8 generatedby usingangles of 27°, 53°, and 90° in Schiaparelli'sscheme instead of the90°, -45°, 90° used by Dreyerand Heathto illustratethe arrangement.Figure 33 showsthe resulting planetary motion using Mars's ratioof synodicto eclipticto synodicperiod (0.88). Retrogrademotion is produced,but it stillfails to reproducethe basic loopedform that characterizes typical Marian retrogressions

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Figure34

Figure35 Figure34 showsthe curve created by three spheres. The polesof the first are fixedto theecliptic (depictedby thedotted great circle that bisects the two smalllatitude circles). The poles of the secondare tilted by 17° relativeto the poles of the first; the poles of the third are tilted by 45° with respectto those of the second, and the planet is placedon latitude25° ofthe third sphere measured relativeto its pole. By startingthe planetat an angle of -.4° on its latituderelative to the ecliptic,the whole resulting curve is slightlyturned with respect to theecliptic. Figure 35 shows thetrace that results when the planet is movedlike Mars around the ecliptic, completing 88% ofits motionaround the curve of Fig. 34 by thetime it completesone revolutionaround the ecliptic. The angleof view in Fig .35 is rotatedto show a view of theretrograde phase as seen by an observerat the center of the stellar sphere. This should be comparedwith the observed retrogression ofMars in Leo fromJune '94 to April'95

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions 264 I. Yavetz planetin this theoretical course will not quite match the motion of Mars. The observation ofsuch discrepancies, however, would require the sort of careful timed observations that arenot evident in Greekastronomy prior to thethird century BC at theearliest. Barring sucherrors, the general shape of the theoretical loop mayquite reasonably be described as havingexcellent qualitative agreement with the observed trace of Mars's retrograde motion.Venus is slightlymore difficult to satisfy,because its synodic period is only5/8 ofits sidereal period, but here also, close likenessto itsactual path may be obtainedby selectingthree larger inclinations for the three inner spheres to create a longerretrograde generatingloop.48 Figures36-53 belowdemonstrate the ability of the alternative Callippic arrangement toreproduce the geometrical forms of individual Martian retrogressions. No attempthas been madeto arriveat a precalculatedbest fit, which would be quiteoutside the most optimisticassessment of analyticalabilities in classicalGreece. The particularparame- tersfor each retrogressionrepresent one outof a rangeof possiblechoices that would satisfya qualitative,visually gauged similarity of form. In all ofthe examples, the eclipticsphere rotates in thesame direction (west to eastrelative to the stellar sphere), with a periodof 0.88 ofthe synodic period, which is thetime it takes for the three inner spheres to completea singlerevolution (recall that relative to an observeron theecliptic sphere, all threeinner spheres appear to spinequally fast; the third and fifthspheres rotate in thesame direction,the fourth spins opposite to them).The mainretrogression features thevarious skew directions of the loops andtheir locations on bothsides of the ecliptic, are all producedby smallvariations of thephase and threeinclination angles, keeping constanttheir rotational directions and speeds of the spheres about their respective poles. All ofthe theoretical paths are traced on theback side ofthe stellar sphere, to presenta view"over the shoulder" of an earthboundobserver at the common center of the spheres. Theyshow only a smallarea of thesky centered on theretrograde loop. In mostcases, in additionto the retrograde loop, the figures contain a non-retrogradetrace from Mars's previouspassage throughthis area. These are irrelevantto thecomparison, and may be ignored.In all cases, theecliptic is markedby a seriesof dotsmore widely spaced thanthose that plot the retrograde trace. The reproductionsof Martian motion based on moderncalculations are plotted approximately in thesame scale as thetheoretical curves producedby the alternative Callippic arrangement. To facilitatecomparison, the dots that markthe ecliptic in theCallippic figures have been plotted every 2° of arc.The angular size of each theoreticalloop maybe guagedagainst this measure, and comparedto the sizes notedbelow each reproductionof Mars's actualmotion. Note that while the forms

48 Use, forexample, angles of 23°, 60°, and 32° forthe inclination between spheres 3 and4, 4 and 5, andthe planet's latitude relative to thepole ofthe. (innermost) sphere 5 respectively,and a phaseangle of 2°. Superimposethe resulting retrograde generating loop on a meanecliptic motion whoseperiod is 5/8of thesynodic period (being the time to go aroundthe retrograde generator). This will resultin a typicallyskewed retrograde loop roughly10° longand 2° wide. Note again thatcomparison of thetheoretical path with direct observation is practicallyimpossible because of Venus'sproximity to thesun.

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions On theHomocentric Spheres of Eudoxus 265 of individualMartian retrogressionscan be approximatedquite well by the alternative Callippic arrangementof fourconcentric spheres, the location of the retrogradetrace relativeto the ecliptic can be up to 3° off.(For furtherdetails on the constructionof the theoreticalpaths, see appendix, §&).

Fig. 36. Marsin Leo, 9/79-7/80;loop size: 19° X 2G6'

••*•*". • * "

Fig. 37. Phaseangle: 0.3°, 1stinclination: 16.75°, 2nd: 44.5°, 3rd:24.75°

To the extentthat reproducing the formof any given planetaryretrogression was the problemto which Eudoxus and Callippus directedtheir efforts, it can be said that if Callippus developed the theoryalong the lines of the alternativereconstruction, then he successfullyfinished what Eudoxus had begun. All of this should not be taken to mean that Callippus solved the problem of planetaryretrograde motion. Indeed, the alternativearrangement can reproducea good likeness of any Martianretrograde loop. However,once fixedwith a given set of specificparameters, the arrangementwill keep

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Fig. 38. Marsin Virgo,10/81-8/82; loop dimensions:18C32' X 0c33'

.••***«• *

Fig. 39. Phase: 1.2°, 1stinclination: 17°, 2nd: 44°, 3rd:24.75° reproducingthe same retrograde loop overand over.The schemecannot automatically reproducethe variationin the formof the retrogressionsfrom one to the next.The inclinationsand phase angle mustbe reassignedfor each new variation,and eventhe rotationaldirections of the three inner spheres have to be reversedin orderto createthe fullrange of Martianretrogressions (the speeds remain constant and alwaysreflect the meansynodic period of theplanet). Absence of variabilityremains a generalfailure of anyEudoxian or Callippictheory. In otherwords, Callippus successfully finished what Eudoxushad begun only to theextent of retrospectively accounting for the geometrical formof any given Martian retrogression, but not to the extent of predicting the variations in formfrom one synodicperiod to thenext. This, however, cannot be ascertainedby randomobservations. One mightcarefully observe Mars in retrogressiononce, then choose appropriateparameters for the spheres to reproducethe motion. Slightly more

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Fig. 40. Marsin Libra,12/83-10/84; zigzag dimensions: 16C45' X 3°56'

^..* #

Fig. 41. Phase:2°, 1stinclination: 17.25°, 2nd: 42.5°, 3rd:25.25°

thantwo years later,another Martian retrogressionmight be observed, with the likely resultof revealingthe inadequacy of the previous choice of parameters.At this point, one mightsimply choose a new set of parameters,only to be frustratedagain by thenext retrogradephase. It takes systematicobservations of Mars in retrogressionfor 15 years at the veryleast, and probablycloser to twice as long, before any sense of the cyclical natureof thesevariations can begin to be perceivedwith any degree of confidence.There mightnot have been sufficienttime for such data to be collected in Callippus's lifetime. However,the very existence of cinematictheories such as Eudoxus and Callippus created could providea powerfulincentive for systematic observations of planetarymotion. That theearliest known records of such observationsin Greece date back to the beginningof the 3rd centuryBC is again in accordance withthe fact thatCallippus flourishedin the second half of the 4th centuryBC.

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Fig. 42. Mars in Sagittarius,3/86-1 1/86; loop dimensions: 1147' X Ie 1 Y

Fig. 43. Phase: 1.4°, 1stinclination: 15.5°, 2nd:38°, 3rd:27°.

Paul Tanneryfound it difficult to believethat with their cinematic theories explicitly formulated,none of Eudoxus's immediate followers was movedto observethe motions ofthe planets. He conjecturedthat no recordof their observations survived because they wereperceived as lectureexperiments designed to demonstratea theory rather than as researchexperiments designed for the critical evaluation of a theory.49Regardless of

49 "On pourraitles comparerà des expériencesde physiqued'amphithéâtre, non pas à des recherchesde laboratoire."Paul Tannery,Recherches Sur L'HistoireDe L'AstronomieAncienne, (Paris:Gauthier- Villars & Fils, 1893,Reprinted by ArnoPress, New York,1976), p. 45. Heath viewedthings differently: "Whether Callippus actually arranged his additional spheres in the way suggestedby Schiaparellior not,the improvements which he madewere doubtless of thenature

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Fig. 44. Marsin Pisces,5/88-2/89; loop dimensions:1 1°47' X 0°19'

__-.~r- V

Fig. 45. Phase: -1.9°, 1stinclination: 16°, 2nd: 38.5°, 3rd: 26°. why certainrecords did or did not survive,there is nothingto preventwhat mighthave startedas a demonstrationexperiment from turning into a criticalcomparison of theory withobservation. Furthermore, Tannery's conjecture notwithstanding, nothing that we know forbidsthe possibilitythat Eudoxus, Callippus, or any of theircontemporaries, purposefullyturned their attention to planetaryretrograde motion for the purpose of assessing the value of theirmost recent theorieson the subject. All of the alternative homocentricreconstructions of Eudoxus and Callippus's workthat have been discussed

indicatedabove; and his motive was thatof better 'saving the phenomena', his comparison of the theoryof Eudoxus with the results of actualobservation having revealed differences sufficiently pronouncedto necessitatea remodelingof the theory." (Heath, Aristarchus of Samos, p. 216).

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Fig. 46. Marsin Taurus,6/90-5/91; zigzag dimensions:16°52' X 3°48'

••***•* *

Fig. 47. Phase: -1.7°, 1stinclination: 17°, 2nd: 43°, 3°: 24.75° in thispaper (Schiaparelli's included) are consistentwith these possibilities. However, theydiffer in thedegree of harmonybetween theory and observationthat they entail. Unfortunately,without additional information it does notseem possible to decidehow Eudoxusand his followersregarded the relationship between astronomical theory and astronomicalobservations.

h. Conclusionsand Conjectures

In general,expositions of theEudoxian system in thecurrent secondary literature reflectHeath's assessment that Schiaparelli's interpretation ". . . willno doubt be accepted by all futurehistorians (in theabsence of thediscovery of freshoriginal documents) as

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Fig.48. Marsin Gemini,8/92-6/93; loop dimensions:18°48' X 1°27'

t.»*• . *

Fig. 49. Phase: -0.8°, 1stinclination: 17°, 2nd: 44.5°, 3°: 24.75° theauthoritative and finalexposition of thesystem."50 This exclusiveidentification of theEudoxian homocentric scheme with Schiaparelli's reconstruction seems unusually specificand ambiguity-freecompared to ouruncertain knowledge of otheraspects of

50 Heath,Aristarchus of Samos,p. 194. An exceptionto thisrule is D.R. Dicks's measured observationthat "Schiaparelli's reconstruction (which is, of course,hypothetical, since the geo- metricaldetails are nowheregiven in theancient sources) has been generallyaccepted, and the accountsof Dreyer and Heathare both closely based on it."(Early Greek Astronomy toAristotle, [London:Thames and Hudson, 1970], p. 177).

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Fig. 50. Marsin Leo, 9/94-8/95;loop dimensions:19°22' X 2 31'

•£****

yy"" * 9e* •

Fig. 51. Phase0.1°, 1stinclination: 17°, 2nd: 45°, 3rd:25° early Greek astronomicalthought. By contrast,this study suggests that unless further evidence can be broughtto bear on the subject, our knowledge of Eudoxian astronomy is not as unusual as mightbe gathered from Schiaparelli's reconstruction.Aristotle's textis equally interpretableeither according to Schiaparelli or according to the general version of the alternativeview described here. Accommodation of both Simplicius's and Aristotle'stexts does call fora furtherrestriction on the alternativeinterpretation of Aristotle's text,but it does not require a full reversionto Schiaparelli's interpretation. Furthermore,there is no a-priorirequirement that all aspects of both texts need to be satisfied.In fact,Schiaparelli himselffound it necessary to correctSimplicius on some pointsof Eudoxus's system.At presentwe have no hardevidence, director indirect,that enables us to make a sound choice betweenthe various alternatives.Such a choice would rely too heavily on an unduly restrictedinterpretation of the single word "hippopede"

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Fig. 52. Marsin Virgo,10/96-8/97; loop dimensions:19° 14' X Io 11'

.***".* *

Fig. 53. Phase:0.9°, 1stinclination: 17.25°, 2nd: 44.75°, 3rd:25°

written900 yearsafter the fact, in a second-handaccount by a writerwho provedhimself less thanreliable in the verysame context.Of course, this does not give license to pick and choose fromSimplicius's textat will withouthistorical justification, but this applies with equal force to Schiaparelli's reconstruction,as well as the new alternativesthat have been developed here.In the finalanalysis, given our presentstate of knowledge,we may only advance the alternativeinterpretations of Aristotle'stestimony as possibilities to be considered alongside Schiaparelli's reconstruction,not as substitutesfor it. With the paucity of informationthat characterizes the historyof Greek astronomyin the 4th centuryBC, it is practicallyimpossible not to have several possible interpretive orientations,and it would appear thatsuspension ofjudgement is thebetter part of valor.

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We mayhave to be contentwith the understanding that spherically symmetrical as- tronomycould have been considerably more flexible in ancientGreece than Simplicius and Schiaparellisuggest. As suchit mayhave encourageda greatdeal of speculative studywithin its framework and in itsterms. Indeed, in itsgeneral Aristotelian version, thisspherical astronomy is capable of supplyingthe basic conceptswith which later thinkerscould breakits homocentricconstraints and lead to the epicycloidalastron- omyof Ptolemy.The basic schemeillustrated in Figs. 4, 5, and 8 clearlyindicates the deferent-epicyclearrangements that Apollonius began to studyand thatcame to char- acterizePtolemaic astronomy. Passage to themodels of Ptolemyrequires extrication of thesesuperimposed rotations from the spherical surface, and embedding them in planes thatare parallel,or nearlyparallel to theplane of theecliptic. This sufficesto create thebasic epicycloidevident in astronomyfrom Ptolemy to Copernicus.The alternative interpretationtherefore suggests the possibility of a pre-existinggeometrical motivation forthis model and hencea naturalconceptual affiliation between Eudoxus and Apollo- nius.The difficultiesencountered by Eudoxus's system would thus not entail a complete overthrowof his geometrical tools, which with a relativelysimple modification may have been foundquite useful outside the general astronomical scheme that he wove around them. Finally,both Schiaparelli's reconstruction and thealternative outlined in thispaper are in keepingwith the presently accepted limitations on thestate of observationalas- tronomyprior to thework of Timocharisin theearly 3rd century BC, butthey suggest differentpossible reconstructions of theevolution of Greekastronomy following Eu- doxus.Under Schiaparelli's interpretation, theEudoxian and Callippic models could not pass thetest of rudimentary observations well within the capacity of Greek astronomers at thetime. That such modelswere advanced just the same seems to indicatea rela- tivelylow level of interestin observationalplanetary astronomy, at leaston thepart of Eudoxusand Callippus. Schiaparelli's interpretation, then, depicts a widegulf between theoryand reality,and suggestsa ratheridealistic development of theorythat was only marginallyguided or correctedby observation.It inevitablyleads to assessmentssuch as thefollowing from Pannekoek: The greatGreek scientists were not observers, not astronomers, but keen thinkers andmathematicians. Eudoxus' theory of the homocentric spheres is memorable notas a lastingacquisition of astronomybut as a monumentof mathematical ingenuity.51 The alternativeEudoxian model leads to a Callippicsystem that has thetheoretical capabilityof reproducinga good likenessof anyplanetary retrograde loop. However, evenif this was thescheme used by Eudoxusand Callippus, it is stillpossible that both theyand theircontemporaries remained unaware of its fullpotential. Indeed, we do noteven knowwhether Eudoxus or Callippusattempted to extractspecific planetary retrogradeloops fromtheir possible arrangements of homocentricspheres, let alone comparethem with actual observations. Consequently, the superior inherent capability

51 A. Pannekoek,A Historyof Astronomy, (London: George Allen & Unwin,LTD., 1961;New York:Dover, 1989), p. 111.

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of thealternative model does notnecessarily imply a finerobservational knowledge of theloops traced by retrogressing planets, and Pannekoek's judgement could hold equally trueunder the alternative interpretation as under Schiaparelli's. However, unlike Schia- parelli'sreconstruction the alternative one also makespossible - thoughnot necessary - a moreharmonious depiction of thecoexistence of theorywith observation. The ob- servationalpractices required for such a coexistenceare significantlyless sophisticated thanthe precise dated observations of the 3rd century BC, butstill sufficient for critically studyingthe primary astronomical phenomena that the theory was presumablydesigned to address,namely, planetary retrograde motion. Besides Pannekoek'sview, then, the alternativeinterpretation also makespossible a moreharmonious and gradual evolution of Greektheoretical and observationalplanetary astronomy from Eudoxus's and Cal- lippus'sstudies of phenomenalocalized in smallareas of thesky, to Ptolemy'sglobal mappingrelative to a standardizedcelestial grid. While our current state of knowledge makesit moreprobable that Eudoxus and Callippusdid notengage in thesort of ob- servationsimplied by this view, we do notpossess sufficient evidence to rejectit outof hand.

Appendix

a. Obtainingan analyticalexpression for the hippopede.

Throughoutthe discussion, the observer's coordinate system is definedwith the z- axispointing northward on thepage, the j-axis pointingeast, and the x-axis pointing up fromthe face of the page. Positive rotations around any axis are defined by the right hand rule.We firstgenerate the rotation operators (this is just forcompleteness of discussion here,they may by foundin almostany text on classicalmechanics). Rotate by anglea roundthe z-axis. This yields:

z = z; xf= x cos a - y sina; yf= x sina + y cos a

Thismay be representedby the matrix operation:

"cosa -sin a 0~| |~jc~ sina cosa O y 0 0 lj '_z_ wherethe 3x3 matrixrepresents the operator that rotates any point round the z-axis. It is easy to showby similarconsiderations that the operators for rotation around the y- andjc-axes are respectively: " 0 0 cosß sinßl |~10 0 1 0 ; 0 cos y - siny _- sinß 0 cosß J Osiny cosy

Usingthese, we createthe hippopede in threesteps: 1) Turna planetresting at (1,0,0)by a roundthe z-axis:

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"cosa -sina 0~| [~1~| Teosa" sina cosa 0 0 = sina o o íj Loj L o 2) Turnthe result by y aroundthe jc-axis to createthe inclination between the poles ofthe outer and innerspheres. This willcomplete the function of theinner sphere: "10 0 "I fcos a "I F cosa 0 cos y - siny sina = cos y sina 0 siny cosy J |_ 0 J L smY sma _ 3) Wenow have to turn the outer sphere, namely the result of the first two operations, by an angle-a aroundthe z-axis: " " " cos a sina 0 cos a cos2a 4- cos y sin2a - sina cosa 0 cosy sina = (cosy - 1) sina cosa (1) 0 0 1 _ _sin y sina _ _ siny sina The rightside of Eq. (1) willgenerate the hippopede when a is allowedto range from 0° to 360°. The size ofthe hippopede is determinedby y. One can see thatthe relationship betweenthe x andz coordinatesis: (cosy- 1) 2 x = H s z , sin y whichdefines a parabolain the x - z plane,with the x coordinateas theaxis of symmetry. If a parabolicsheet is createdby extrudingthis parabola along the y-axis,then the hippopedeis formedby the intersection of thissheet and a sphere.

b. Derivationof the trace lefton a sphereby thealternative scheme

Considera planetfixed on latitude8 (measuredfrom the pole) of a sphere.Fix this sphereinside another sphere, whose own pole is inclinedto thepole ofthe planet carrier by an angley. Spin thetwo spheres in oppositedirections at equal speedsand observe theplanet's motion. It is equivalentto thecombined motion of an epicycleof angular size 28 on a deferentof angularsize 2y lyingon a sphericalsurface, and turningin oppositesenses at equal speeds.The basic procedureis to tiltthe vector (1, 0, 0) by 8 aroundthe j-axis, thenturn it 2a roundthe jc-axis, then tilt the whole thing a further y roundthe j-axis, and finallyturn by -a aroundthe Jt-axis. The outcomeresults in they-tilted center of 2a rotationbeing rotated around the jc-axis by -a, whilethe net rotationaround this tilted pole is reducedto a. So we followfour distinct steps: 1) Rotatethe vector (1, 0, 0) by8 roundthe j-axis, tocreate the rotating radius vector thatwill create the epicycle: cos 8 0 sin5 ~| F 1 ~j F cos 8 0 10 0 = 0 _- sin<50cos<$J [_0j L~sm^-

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions Onthe Homocentric Spheres of Eudoxus 277

2) Turnthis by 2a roundthe x-axis to startthe epicyclic component: "10 0 "I F cos<5 "I I" cos 8 0 cos 2a - sin2a 0 = sin2a sin8 0 sin2a cos 2a J |_- sin5 J [_- cos 2a sin5 _ (Note thatrotating twice by a givesthe same result,as it mustif theoperators are to representrotations properly). 3) Tiltby y roundthe y -axis to place the epicycle on thecircumference ofa deferent: " cos y 0 siny "I cos 8 cos y cos 8 - siny cos 2a sin8 0 10 sin2a sin8 = sin2a sin8 _- siny 0 cos y J - cos 2a sin8 _ _- siny cos 8 - cos y cos 2a sin8 _ 4) Turnby -a aroundthe jc-axis to move the center of the epicycle along the deferent, andreduce the epicycle's rotation to a: - "1 0 0 ~| F cos y cos 8 siny cos 2a sin8 0 cosa sina sin2a sin8 = 0 - sina cos a J [_- siny cos 8 - cos y cos 2a sin8 _

- cos y cos 8 siny cos 2a sin8 ~| cos a sin2a sin8 - sina siny cos <5- sina cos y cos 2a sin8 - - _ sina sin2a sin8 cos a siny cos 8 - cos a cos y cos 2a sin<5 J Thisyields the observer's coordinates for the resulting motion. Moregenerally, the position of a pointon a givenlatitude 0 measuredrelative to the x-axisof a spherethat rotates a degreesabout the jc-axis may be specifiedby two matrix operatorsas follows: "10 0 "I I" coso 0 sino"! [~1~] |~jc" 0 cos a- sina 0 10 0 = y 0 sina cosa J |_- sino0 cosoJ |_0j |_£_ To superimposeanother spherical rotation on theabove, apply another pair of matrix operatorsto the vector (x, y, z) witha newangle of inclination and a newrotation. This maybe repeatedany number of times, with different inclinations and different rotations. Inthe case ofCallippus, we needthree such operator pairs. The three angles of inclination mustbe chosenso as to createa retrograde-generatingcurve of theappropriate length andwidth, while the rotational angles must be a forthe third sphere, -2a forthe fourth, and 2a forthe fifth. Throughout the operation, the coordinate system is fixed,and only thevectors are rotated. The coordinatesystem is chosenso thatthe x-axis points directly outwardfrom the screen, the j-axis pointshorizontally to the right of the screen, and the z-axispoints vertically up. The order, symbolically, is: [a+0]x[0]y [-2a]x[<5]y[2a]x[)/]y ex. Hereexis a unitvector in the x direction,a is therunning variable, 0 thephase angle (theangle between the long axis of theretrograde generator and theecliptic), y is the latitude(measured from the pole, notthe equator) of theplanet on theinnermost (5th overall)sphere, 8 is theinclination of the 5th sphere relative to the 4th, 0 is theinclination ofthe 4threlative to the3rd, and the poles ofthe third sphere, as always,are fixedto the

This content downloaded from 128.196.132.173 on Fri, 26 Sep 2014 22:57:27 PM All use subject to JSTOR Terms and Conditions 278 I. Yavetz ecliptic,being the equator of the 2nd sphere. The subscriptsoutside the brackets signify theaxis of rotation,so that[y]y ex signifiesa counter-clockwiserotation of (1,0,0) by y aroundthe y-axis. To completethe picture, one morerotation around the y-axis is requiredto createthe mean ecliptic motion of theplanet, say [ß]y. Both ß and a are runningvariables, such thatAa/Aß= (synodicangular speed)/(mean ecliptic angular speed) = (mean eclipticperiod)/( synodic period) for the planetunder consideration (about0.88 in thecase of Mars). For instance,the curve in Fig. 34 was createdby placingthe planet on latitude25° of theinnermost sphere, which rotates about its axis at a speed of 2a. The innermost sphere'saxis is inclinedby 45° tothe axis ofthe sphere above it, which rotates at a speed of -2a, and whichis itselfinclined by 17° relativeto theaxis ofthe next higher sphere whichrotates at a speedof a. Thislast axis is attachedto theecliptic, and carried around it at themean ecliptic period of theplanet. To see howthis creates the resulting curve, considerfirst that the axis ofthe innermost sphere is movingon thealternative Eudoxian curvecreated by placing a planeton latitude45° ofthe Eudoxian 4th sphere whose axis is inclinedby 17° relativeto theEudoxian 3rd sphere. This createsa curvewith a 62° semi-majoraxis, and a 28°semi-minor. In theCallippean scheme, the planet is rotating on latitude25° measuredfrom a pole thatfollows the Eudoxian path. This resultsin a curvewith an 87° semi-majoraxis, and a 3° semi-minoraxis, which is reproducedin Fig. 34. In thisfigure the retrograde loop is situatedto theleft of theecliptic. Planets sometimesretrogress to the left, and sometimes to the right of the ecliptic. Reflecting the retrogradeloop in Fig. 35 to theright of theecliptic requires reversing the direction of motionalong the retrogression generator in Fig. 34. To obtainthis effect, use inclinations of 20°, 42°, and 25°. Thisplaces theplanet on latitude25° of a spinningsphere whose pole travelsalong a Eudoxiangenerator with a same semi-majoraxis of 62° as before, and a semi-minoraxis of 22°, as opposedto 28° before.Because theplanet's latitude on theinnermost sphere is now 3° largerthan the difference between the 1stand 2nd inclinationsinstead of 3° smaller,the left and rightsides of theresulting retrogression generatorare switched, which has the effect of reversing the direction of motion. As Figs. 6 and 7 show,the reversed generator is notexactly congruent with the generator in Fig. 34, despitehaving the same semi-major and semi-minoraxes (87° and 3° respectively). However,since the angular size differencesbetween the inclinations that create the two versionsare small,the divergence of formis negligiblein applicationto thecreation of Martianretrograde loops at therelevant level of accuracy.52 Institutefor the History and PhilosophyofScience and Ideas TelAviv University RamatAviv Israel69978

(ReceivedSeptember 7, 7997)

52 An earlyversion of thispaper was readat theDibner Institute's Colloquium in thecourse of a Dibnerfellowship for 1 996-97, during which much of the work on thepaper was carriedout. I thankNoel Swerdlowfor several useful suggestions and criticalremarks.

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