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

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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 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Springer is collaborating with JSTOR to digitize, preserve and extend access to Archive for History of Exact Sciences. http://www.jstor.org 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 Arch.Hist. Exact Sci. 52 (1998) 221-278. © Springer-Verlag 1998 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 reconstruc- tionof 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. 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 222 I. Yavetz 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). 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 223 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 sur- face 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. 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 224 I. Yavetz 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 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 225 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
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