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Astronomical & Astrophysical Transactions The Journal of the Eurasian Astronomical Society Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713453505 Comparison and historical evolution of ancient Greek cosmological ideas and mathematical models Antonios D. Pinotsis a a Section of Astrophysics, Astronomy and Mechanics, Department of Physics, University of Athens, Athens, Greece

Online Publication Date: 01 December 2005 To cite this Article: Pinotsis, Antonios D. (2005) 'Comparison and historical evolution of ancient Greek cosmological ideas and mathematical models', Astronomical & Astrophysical Transactions, 24:6, 463 - 483 To link to this article: DOI: 10.1080/10556790600603859 URL: http://dx.doi.org/10.1080/10556790600603859

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System. century Solar fifth the late of the centre in the as Earth did the Pythagoreans consider some not system.’However, geocentric the accepted Democritus and Leucippus however, Others, manner. circular Syracusian.’ also a the in He moves Hicetas Earth need. to the it a that find attribute of to first because the and was he harmoniously that believed happens everything that believed Hicetas. ‘[Philolaus] of 39, ideas (II, the centre Cicero the for in axis.’Also, same being central the as its mentioned it around [27]) regard moving of others and 123 and (whirling) stars Timaeus eddying the Universe, in of the mentioned one of as is Earth it the is as characterize think Universe some do the said, they of What axis sphere. celestial the the around of moves centre the the in lies found there is that Earth the that think not did ihrgr oteieso yhgras ly p.27o 4)ntd ‘ noted: [4]) of 267 (pp. Lloyd Pythagoreans, of ideas the to regard With hr sa bigmn fDoee aris(,8 f[] hc adtefollowing: the said which [9]) of 85 (8, Laertius Diogenes of abridgement an is There tcnb eue rmti rgetta lt eivdta iewsbr simultaneously born was time that believed Plato that fragment this from deduced be can It utemr,mn tessaeteble fteiainPioohr,ta esol not should we that Philosophers, italian the of belief the share others many Furthermore, n h at ic ti n ftesasadrttsaon h etecetsteday the creates centre the around rotates and stars the of one is it since Earth the and Timaeus Plato 3b f[28]) of c (38b, Fire . nin re omlgcliesadmteaia models mathematical and ideas cosmological Greek Ancient ... oeee a htteErhb cuyn h eteoclae and oscillates centre the occupying by Earth the that say even Some † Tm a hrfr onaogwt smlaeul)tesyin sky the (simultaneously) with along born therefore was ‘Time : ... hrfr nta a ftikn they thinking of way that in therefore ... sw aealready have we As ... ti ieythat likely is it 467 Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 ntecnr fteworld). the of centre the in oin ftepaescudb ecie i h opsto fnra uiom circular them. 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Sun The each the in motion. to period corresponding monthly different spheres its a of namely with subset but each circle, it planets of zodiac that five sphere the conclude second may the along to One Moon The applies texts. the ecliptic. same ancient the of in of virtually motion clearly plane was explained middle the sphere not the second describes is to the sphere perpendicular of this was equator of rotation the way, significance of that axis In The circle. first. zodiac the the to way motion opposite daily an the in describe would planet. which each spheres, and of Sun subset the each Moon, performed was the of motion of the sphere same of (outer) the period Exactly first motion. rotational the celestial The daily by axis equator. the The celestial depicting poles. the h, world 24 of the was plane sphere through the go to would perpendicular turn therefore in was which axis, its around westwards rotating h uou hoyadepcal h trbtso h gr-fegtcurve, figure-of-eight explain the to of tried attributes Neugebauer books. the Schia- their of especially work in and the Eudoxus theory on of Eudoxus (especially theory studies the the these describe on to based 197 order were 196, in [33]) why (pp. of parelli) Neugebauer 677 reason pp. [14], the [32]; Dicks of is [13], 228 Dreyer that and [8], and Heath sources system. Eudoxus’ ancient of these principles obtained in geometrical who mentioned The [22]), not Sosigenes. of are philosopher more theory 221a a the that mainly and through of 31]) 12, Rhodes [30, details (II, also of Simplicius see Eudemus [29]; by from of book 1073b information the 8, (XI, in Aristotle description by book detailed the in found are theory ii h w ns(h oe)o h xso ahshr nec usto pee would spheres of subset each in sphere each of axis the of poles) (the ends two The (iii) the and stars. revolving Moon fixed the 27 the for for of one three set and spheres: a planet of with each subsets for him eight four before Sun, of formulated consisted been which had spheres, that concentric orbits circular the replaced h reuaiiso hi rish omltdteter fhmcnrcshrs hc he which spheres, homocentric of theory the formulated book he his in geo- orbits and described first bodies their celestial the the invented of of he motions irregularities apparent Plato, the the by explain to Motivated order immobile. In model. was cosmological Earth metric the that Plato, teacher his 2.6 i)Teshrswudrtt neednl n rmteohr ahwt t w constant own its with each other, the from one independently rotate would spheres The (ii) i h pee eentpretbefo h ua essadtesilshrclErhwas Earth spherical still the and senses human the from perceptible not were spheres The (i) codn ota hoy h uems pee pnwihtefie tr eelig was lying, were stars fixed the which upon sphere, outermost the theory, that to According h hr pee nteeutro hc h onwsfud ol oaewestwards rotate would found, was Moon the which of equator the on sphere, third The h ai rnilso i ytmwr h following. the were system his of principles basic The ntesbe fshrscrepnigt h on h eodshr a oaigeastwards rotating was sphere second the Moon, the to corresponding spheres of subset the In nodrt xli h paetmtoso h on h u n h v lnt,Eudoxus planets, five the and Sun the Moon, the of motions apparent the explain to order In civdtruha prpit obnto fmtoso h pee.W observe We spheres. of the principle of be the motions perceived therefore motions already could of had body Eudoxus combination celestial that each appropriate of an motion through the achieved of description The it. surrounding peetkspr ntemto ftepeeigspheres, preceding the each of Thus, sphere. motion preceding the immediately in the part of surface takes internal sphere the on fixed firmly be the through pass would Earth. which the axis, of its centre around motion) circular (normal velocity angular centre. common sphere’s) (the their on located uou fCnidus of Eudoxus oee,w aeas nonee hs da nPaosvesin views Plato’s in ideas these encountered also have we However, . nVelocities On nin re omlgcliesadmteaia models mathematical and ideas cosmological Greek Ancient (ερ ιTαχ ´ ων) ˜ hsbo a enls.Rfrne ohis to References lost. been has book This . needneadsproiinof superposition and independence i.e. h pee hc were which spheres the hippopede Timaeus . . 469 Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 poiedrcin htcrewsnmdatrEdxshippopede towards Eudoxus but after period named same was the curve with oscillations That Namely, harmonic direction. it. two opposite by of bisected superposition was the which have and ecliptic we the here along was which the curve, of figure-of-eight declinations a out breadthwise the as well interpret as could The motions planet planet. retrograde an each the of for exhibit spheres and different two points be would however, last stationary the that would, axis the of which to rotational motions sphere, of equal third Its superposition period the The direction. planet. of a each axis opposite at the the rotate to would towards respect with found, but fourth inclination was the sphere, Finally, planet third planet. towards the each the place which of of taking of rotation synodic was equator the the the Rotation On second to Venus. on planet. equal and the sphere, period each Mercury a of for in for different equator direction same were direct the fixed the the were meaning were poles of they ecliptic, two hand, which period the the axis, other zodiac which on its on or around fixed points rotate stellar constantly The would the sphere. sphere poles third to its The equal have plane. period ecliptic would, was the a which on with that moved sphere, planet, second direction overcome the The direct to planet. the order each towards for In sphere rotating motion. extra one retrograde adopted and Eudoxus difficulty, (stagnations) points stationary the solstices. interpret motions, the not explain could not it could since it inadequate, example for was motions; theory observed That already possessed. Eudoxus Sun which its the the ecliptic) and the that to sphere from second believed (declination respect the motion than breadthwise with slowly the interpret more inclination to rotate was small would purpose sphere a second That sphere. the had second like which the rotate, of would axis, axis would sphere sphere its (third) second internal around the The of year. counterclockwise rotation 1 sphere, the of case period this a in with slow, where fairly Moon, be the corresponding of motions spheres perform first therefore would two spheres the two to first The Moon. the of that to gous modern by formulated been have sphere uncertainties third these the the authors. concerning was of views motion which Opposing the or years. the Probably 18.6 sphere abnormalities, sphere. Moon’s third third the the of the one and of described axis second value rotational the which the certainty for with of known adopted inclination not Eudoxus is It participate velocity it. would surrounded angular in sphere that changes of spheres that the two depict mentioned, the would of already it motions have that the of we in possible what breadthwise quite is to motion It According reciprocating ecliptic). declination. the the its of explain sides to both aimed (on Moon sphere the That sphere. second the of 470 fPtn aot30B) u o ucsfly(p 0 f[14]). of 306 (pp. successfully not but BC), 300 (about Pitane of could Eudoxus of the phenomena. spheres with observed concentric those The together too. explain indicates Moon changing therefore this was the not and Earth of changing the however, from diameter were, distance Venus apparent their and that The Mercury Earth. Mars, of moves the body brightness around celestial apparent the orbit as circular long as a constant, stay on should brightness the model. and his diameter with are apparent details hippopede numerical the on associate motion not the did planet by Eudoxus produced each the latitudes and of the negligible that retrogradations almost means However, actual This axes. the large. inclined very that not two assumed are about Neugebauer spheres velocity concentric calculations, angular two some of opposite motion after but the constant by with generated is rotate hippopede which the that agree sources all that the as known codn oteter fEdxs h itneo h onado ahpae,the planet, each of and Moon the of distance the Eudoxus, of theory the to According hspolmapae al.h rtwotidt vroeteedfcliswasAutolycus difficulties these overcome to tried who first early.The appeared problem This eie hi ratws elnto h lnt xii,drn hi paetdirect apparent their during exhibit, planets the declination breadthwise their Besides analo- mechanism a considered Eudoxus Sun, the of motion apparent the explain to order In resultant lemniscus oino h pee orsod oa silto ftepae,tracing planet, the of oscillation an to corresponds spheres the of motion sallk)(gr ) loNueae p.67o 3] mentioned [33]) of 677 (pp. Neugebauer Also 1). (figure lake) (small .D Pinotsis D. A. ersino h nodes the of regression (
ππ hc a eidof period a has which o π εδη ´ ) n salso is and Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 rsol nhslt er obe h hoyo uou,bcueh a o totally not was he because Eudoxus, of theory the doubted years late his in Aristotle h oe fEdxswsaotdadetne yhssuetteatooe Callippus astronomer the student his by extended and adopted was Eudoxus of model The foede o a ufiin teto,t hn htPtaoen eetefis who of that, first possible the Geminus is were by it Pythagoreans book which that the to think according in to [35]), attention, some fragment also sufficient or a at see pay Aristotle is discussed [34]; Plato, not been of There by does have 21 mentioned Plato. one was 19, should been if of circles I, it have eccentric partner (book otherwise could of and Rhodes or it since brightness epicycles day, least student of and Simpli- at that idea other distances of eccentricity or to the that in Academy of hand, up with changes Plato’s other idea known coincides the the probably the Theon explain On of not mentioned circles bodies. Theon testimony already celestial eccentric to The had the According planets. the Aristotle phenomena. the because [7]), the of cius, of explain orbits the to 189 about sufficiency (pp. its Smyrna regarding of it by satisfied book his in Aristotle BC). (370–300 2.7 tr a o enadd(I ,17b17ao [29]). fixed of the 1073b–1074a of 8, sphere concentric (XI, the 56 numbers added those been adopted Aristotle In not [15]). He suggested. has of had modified. Eudoxus stars 22, that he 296b, (XIV, 26 which the world a of Eudoxus the instead is of Earth of spheres there centre model the the that cosmological around at concluded the move still he remain Universeadopted bodies to the arguments, celestial that Earth several facts, the the developing other for that among After need and accepted, orbits. He spherical circular [15]). is of normal 14 Earth on 13, the 11, that 9, 8, spherical, II, is and 7, 6, 5, book his I, in in BC) (384–322 byAristotle also supported were theories Similar 2.8 33 complicated. to more (27) same spheres 26 the concentric from retained of Eudoxus he model of the the Saturn spheres and making and the Moon (34), Jupiter increased the therefore For both Callippus Mars. for Eudoxus. and spheres Venus as Mercury, unstellar spheres positions for two apparent added each Sun, the one the and better and Sun approach Moon to the order planets, in the of Callippus, that concisely mentioned [33]) codn ocransources, certain to According iue1 h eutn oino h hr n orhshrs h ippd fEdxso Cnidus. of Eudoxus of hippopede the spheres, fourth and third the of motion resultant The 1. Figure Aristotle Callippus nin re omlgcliesadmteaia models mathematical and ideas cosmological Greek Ancient e.g. Metaphysica ipiis(I 2 2a p 0 n 0 f[22]) of 505 and 504 pp. 226a, 12, (II, Simplicius X,8 03 f[9)(e lop.63of 683 pp. also (see [29]) of 1073b 8, (XI, nHeaven On (mainly 471 Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 rsacu fSms(1–3 C,wowsa srnmradmteaiin formulated mathematician, and astronomer an was who BC), (310–230 Samos of Aristarchus rsacu,wobleeta hyepantepeoeai hycnie h eeta sphere celestial the consider they if phenomena the explain they that believe who Aristarchus, otmoayoAitteadprnro hAaeywsasueto lt n philosopher and Plato of student a was theAcademy of partner and ofAristotle contemporary A h paetmto fec lntwsacmiaino h oino h at n t own its and Earth the of motion the of combination a was planet each of motion apparent The n h olwn I – f[7;seas 3]:‘rsacu fSmshspbihdsome published has Samos of ‘Aristarchus [38]): also see [37]; of 4–6 (I, following the find world. still. the were of stars centre fixed the The of at Sun. idea the remained the around still system Sun that the since In rejected, system. was heliocentric immobility the Earth’s namely the model, cosmological different a hand [in other still the is on Pythagorian Earth the the Philolaus that world]. believe the hand of centre one the on [philosophers] ‘Others accordingly: mentioned 2.10 of poles the around direction direct towards rotates Earth day’. the a that once and equator still the be to stars axis.’ the own and (II, its Simplicius ‘ around of [22]): extracts wheel of some a and 444, Proclus like p. of direction, work 2000b, the 7, direct in towards found be axis can testimonies its orbit, Relevant an around within transitionally rotating not moves, by Earth but the that believe Pythagorian the Ecphantus and year 1 of period a in centre its as Earth the ( having circle a of counterclockwise circumference enn oehn iia owa h yhgrasEpatsadHctshdformulated world, had the ( Hicetas of Plutarch axis and [36]). the Ecphantus of is 316–317 Pythagorians which pp. the axis, (i, its what around to Earth similar the of something rotation meaning direct the to due was Earth the by influenced was Ptolemy that clarify inventor we the 3.3 Heraclides. was section of Heraclides in theory that Later, say epicycles. of essentially system therefore the can We of planets. two the of motions fteSnt h te,rtt ncrua risaon h u,wihi on ttercentre, their at found is which Sun, ( the velocities around side angular orbits one different from sometimes circular but oscillate on to which constant seem rotate with planets), so other, and the internal Sun to the (the behind Sun 255–260 Venus times the other and (pp. of at and Mercury theory of front [36]), original in of be and to 316–317 bright seemed pp. his i, to [8]; According Cicero. of of and Capella Martianus atal omltdahloeti hoy etmne fhssse r on ntework the in found are translated system who Chalcidius his philosopher of neoplatonic Testimonies the theory. of heliocentric a of formulated extract partially had from he apart though today Even survived other. has the work. nothing His on his subjects, Plato of Plato. been different some and had of on hand he Academy books that one ideas the influences many the innovative spiritual in written on great and Pythagoreans two debate strange the the a to his to, extent, develop subjected of great to a because to enough [9]) due, was was of paradoxology presence 72 his (8, therefore spirit. paradoxologist his and of a breadth the called for was distinguished was He who BC), (388–315 Pontus of Heraclides called netdecnrccrlsadeiyls u htde o emt etu p.29o [8]; of 269 2.9 (pp. true be to seem not does [13]). that of but 143 epicycles, pp. and circles eccentric invented 472 eeetcircle deferent ae ntecrosbo fAciee 2722BC) (287–212 Archimedes of book curious the on Based eas dpe h iwta h al ergaertto fteclsilshr rudthe around sphere celestial the of rotation retrograde daily the that view the adopted also He ipoigteter fEdxs oehrwt h ecpin fPaoadAitte he Aristotle, and Plato of perceptions the with together Eudoxus, of theory the Disproving ealdso Pontus of Heraclides rsacu fSamos of Aristarchus or deferent sefiue3ltr) ecneunl xlie rtteapparent the first explained consequently He later)). 3 figure (see ... o hr r oe mn hmHrcie fPnu and Pontus of Heraclides whom among some, are there for  3o 1],a ehv led etoe scin2.3), (section mentioned already have we as [11]), of 13 , .D Pinotsis D. A. epicycles .Smlaeul,teSndpcsthe depicts Sun the Simultaneously, ). Timaeus Psammites noLtn fVtuis of Vitruvius, of Latin, into ... 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Pergi’s order and of in Apollonius circles of quickly figment eccentric very of abandoned system was Copernican, the called approved later was that Aristarchus] [of of mathe- Copernicus a the The way. for Samos, praise the reserved of all of was Aristarchus of step achievement named antiquity.’ final worthy Archimedes, memorable that is with even that cover It contemporary masters to to planets. matician, become corresponds time, the of to which of period so order honor short motion in great had a heliocentric do after Greeks the managed, to the they of them [helio- that concept time for direction general Plato’s much that the wasn’t During of towards there impressively. movement that quite advanced, the fortified enormously Pythagorian], was the system] Philolaus centric means [he him ytmocpe sawoe ob on ncmaio ihtesz fteUies.Ti is This Universe. the of size the with comparison today. in adopt point planetary also a the scientists be and what Earth to the whole, of a orbit as the occupied which system space, the is considered stars surface.’ He fixed Universe. its spherical the to the of sphere distance the of the radius centre to the the the revolution of of of its ratio sphere ratio during the The the that depicts to orbit. big, Earth Earth’s so equal on the is the moves which Sun, of Earth the circle centre with the the centre the that of same in and the located still possesses remain is which to Sun stars, which said fixed the sun, is and the it stars around than fixed circle greater the much a that is assumes world also the He that considers be. he which from [scriptures], theories twsrvvdb oencshmef i li otetteo h nin oencsi still, is Copernicus ancient the of title the unshaken, until to quite again claim opinion, abandoned my His then in himself. was Copernicus which by hypothesis, revived Copernican was the it formulate to first the been affected who discovery. Aristarchus’ Aristarchus, of to contributed predecessors and the were thinking Pontus his of Heraclides Hic- Philolaus, and Ecphantus ofAristarchus.Anaximander, etas, time mathematical as the ideas until peculiar philosophical knowledge and mathematical a astronomical of of as development as gradual well suddenly, a of appear result the not was it did but model, Aristarchus by invented system centric the found mentioned be Cicero. [16]) also and of can Empiricus 1 Aristarchus [20]) Sextus of (H, Vitruvius, of by system book 24.8 books heliocentric another (II, the the in in Aetius to Plutarch disc.’ References following, solar Aristarchus. the the of in the of ideas [near see eclipse node shall the a we near observe is as we Moon and, the orbit] when Earth’s and the Sun of the around plane moves Earth the still, remain stars lo ly p.28o 4)i i okwoe Aitrhster ftehloeti system heliocentric the of ‘Aristarchus’theory wrote: book his in [4]) of 268 (pp. Lloyd Also, h itra fmteais oi Vl ,prgah6,p.14o 3] rt:‘After wrote: [39]) of 114 pp. 65, paragraph I, (Vol. Loria mathematics, of historian The nadto,Lra(o.I aarp 6 p 1 f[9)si taohrpit Tesystem ‘The point: another at said [39]) of 115 pp. 66, paragraph I, (Vol. Loria addition, In ltrh(,K (B, Plutarch of size of concept the grasped had Aristarchus that conclude we analogy, previous the From et i,p.2o 3] omne:‘oAitrhsblnstehg ooro having of honour high the belongs Aristarchus ‘To commented: [36]) of 2 pp. (ii, Heath helio- novel the that conclude we sections previous the in presented arguments the on Based  f[1)si:‘rsacu omltdteter htteSnadtefixed the and Sun the that theory the formulated ‘Aristarchus said: [11]) of nin re omlgcliesadmteaia models mathematical and ideas cosmological Greek Ancient ... .’ 473 Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 fe rsacu fSms eecs(eodcnuyB) otmoayo iprhs an Hipparchus, of contemporary a BC), century (second Seleucus Samos, of Aristarchus After 474 peddrn h hr n eodcnuisB.Temdl feiylsado eccentric of geocentric the and support epicycles to order of in models developed were were The and hypothesis. and BC. spheres models concentric centuries geometrical the purely time second were on the and cycles theory at third his introduced the of first formulation were during the circles spread eccentric after of Cnidus and of epicycles Eudoxus of of ideas the that appears It 3.1 circles eccentric and epicycles of theory The 3. world. known then) (until the about perceptions adverse because and conflicting, different totally two were from theories two arose These they heliocentric. the motion and geocentric and the rotation lated: its to due that Moon.’ claims the well, of as rotation Seleucus moving] the is that impedes Earth it mentioned the also that Plutarch [saying Earth book, the ( another Aristarchus In of mentioned theories Empiricus. is forward, the Sextus same moving followed The of and on.’ scripts later rotating [mediated] the Earth] showed [the in had imagines Seleucus Aristarchus. and ‘He of Aristarchus [16]): as ideas of exactly the 1 supported (H, strongly commented geographer, Plutarch a and mathematician a astronomer, by 2.11 gravity of Kepler. law of the laws of empirical discovery of three explanation the the the and proved 1610, who parallax, 1686, in stellar in Galileo the Newton by and telescope starlight the of heliocentric of aberration the invention of the prevalence the the follows: to as led that were events theory The Copernicus. in of remained time Aristarchus the mainly until and obscurity Heraclides Ecphantus, revolu- Hicetas, The instantly. Philolaus, prevail of not ideas did Unfortunately, tionary science. for horizons new up opening were i)Temto ihntesse feiylsi losmlrt h rvosmto and motion previous the to similar also is epicycles of system the within motion The (ii) i codn oteter fteecnrccrls eeta oyH nteoehn,moves hand, one the on H, body celestial a circles, eccentric the of theory the to According (i) h da fAitrhswr leigteetrl fhmntikn uigta ieand time that during thinking human of entirely the altering were Aristarchus of ideas The ec,utlHpacu’tm 1010B) w ytm a endvlpdadformu- and developed been had systems two BC), (190–120 time Hipparchus’ until Hence, ntecrufrneo icewt eteKadaradius a and K centre a with motion) circle circular a (normal velocity of angular circumference constant the a on with direction) direct the (towards ihacntn nua velocity angular constant a with centre The counterclockwise. otemto faclsilbd na lis,oefcso hc socpe ythe by occupied is which ellipse, of of theory focus the one to ellipse, according is, an motion on the body of eccentricity celestial The a Earth. of motion the to nua eoiyaon h pgeA n ihraglrvlct ntergo of region the in velocity angular higher lower a a has and body A, the particular, apogee perigee In the the circle. different around eccentric a from the velocity with distance of secondly, angular different point and, a each brightness at at its firstly, velocity in moving: angular changes is the body explains celestial which the observer, way eccentricity the this with circle In This body. 2). the (figure of circle observer the of the centre to the considered from is which Earth, the hand, osiue etrapoiain sal,i hscs,aclsilbd oe iha with moves H body celestial a case, velocity this angular constant in Usually, approximation. better a constitutes ploiso eg;Hrcle fPontus of Heraclldes Perga; of Apollonius Seleucus  fiue2.Ti oini setal qiaet safis approximation, first a as equivalent, essentially is motion This 2). (figure ω 2 nasalcrl fradius of circle small a on B ω fteeiyl oe ycneto counterclockwise, convention by moves epicycle the of  1 Zo 1]:‘eecstemteaiin ymoving by mathematician, the ‘Seleucus [11]): of IZ , uulydfeetfrom different (usually .D Pinotsis D. A.  r spae tadsac K distance a at placed is , the , ω epicycle 2 ntecrufrneo a of circumference the on ) R e = ihrcokieor clockwise either , steaprn orbit apparent the is A nteother the On KA. e = K  / = KA. eR Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 ri fHis H of orbit seuvln oteecnrccrl dse ice ihKa t eteadardu H h itnebtenthe between distance The KH. radius a and centre its as K with circle) (dashed circle Earth eccentric the to equivalent is iue2 h ceti oin hc spromdb eeta oyH h at spae tpoint at placed is Earth The H. body celestial a by performed is which motion, eccentric The 2. 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It the harmonic [40]). of simple of plane two 7.2 the (section of with superposition angle a an is at which be may epicycle the of radius a with circle great terrtoms eartoa ubr,tersligmto ilb nelpe(rahelical a (or ellipse way an suitable be will a motion in resulting the etc.). combined number), curve, are rational frequencies a be angular must two ratio (their those that provided general, In  / n h eteKi qa oteradius the to equal is K centre the and A fHi h u,pit n E and E points Sun, the is H If KA. e = K /R = r/R nin re omlgcliesadmteaia models mathematical and ideas cosmological Greek Ancient . R h eeet hc a h at siscentre its as Earth the has which deferent, the , deferent r umradwne osie n points and solstice; winter and summer are r( = BH ihradius with ) fteeiyl.Teeoe h cetiiyo h apparent the of eccentricity the Therefore, epicycle. the of  steprgeo h ceti motion. eccentric the of perigee the is R =  .I steSnte h pcci motion epicyclic the then Sun the is H If B. e = r/R fiue3 p 6 f[33]). of 264 (p. 3) (figure γ and γ  h pigand spring the h plane The .  n its and 475 Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 ploiswstefis h nrdcdecnrccrls hsasmto locicdswith coincides also assumption This circles. eccentric introduced who first the was Apollonius eito hti usd h icmeec fcrl fradius of circle of circumference the outside is that deviation rssi h oino aelt hc osrudteErhi tbecrua ri fradius of orbit circular stable a in Earth the round goes which satellite r a of motion the in arises 476 iue4 h ri hw sadse icei h pcci ri prubdobt hc sams h circumference the almost is which orbit) (perturbed orbit epicyclic the O is circle centre dashed a the as shown however, orbit The circle; of 4. Figure sw aearaymnind h oino necnrccrl s safis approximation, eccentricity first with foci a its as of is, one possesses circle since, Earth eccentric the book) where an relevant ellipse, a on an of motion on motion author the a the mentioned, is that already (he have therefore sections we obvious conic as is introduced Apollonius It that circles. fact cases the eccentric the the studied also and He epicycles apparent planets. their interior the the extent between for certain equivalence implemented a be of to not interprets could it it since while Saturn, motions, and applied Jupiter Mars, first planets BC) exterior in (265–190 mentioned Perga we of as Apollonius Heraclides, geometrician was epicycles great circles. eccentric the the implemented while who 2.9, first section the that seems radial 4). greater it (figure of circle direction eccentric the to an shifted is is motion but resultant Earth) the the Hence, of deviation. centre the (i.e. orbit circular initial fiue2). (figure E nua oetm h aelt ilte xct safis prxmto ipeharmonic simple a position approximation first the a around as execute oscillation then will satellite The momentum. angular where o h w amncmtosi ainlnme n,i particular, in and, number periods) rational corresponding a the is of motions Earth’s ratio harmonic the the two of (or the case velocities for the angular In the orbit. of initial superposition ratio its the the slightly field, of gravitational changes satellite result the a oscillations, two As these oscillations. of harmonic two simultaneously execute therefore eainta h eutn oini e iclrobtsitdfo h nta ri n of and orbit this initial from the deduced from shifted be O orbit can centre circular It The new [40]). period. a of is same 284 motion the resultant p. the also that (see relation calculations) mathematical the omit o nua velocity angular , ploissuidteter fecnrccrlsadpoe hti ple nyt h three the to only applies it that proved and circles eccentric of theory the concerned, studied Apollonius is circles eccentric the and epicycles of system the of inventor the as far As hc eut nacag nisrdu by radius its in change a in results which , D stewdho silto ihaglrfrequency angular with oscillation of width the is ω o n period and ssitdfo h eteOo h at ntedrcino h aiu radial maximum the of direction the in Earth the of O centre the from shifted is r ftenwobtde o oniewt h eteOo the of O centre the with coincide not does orbit new the of o feulbimadbtentelimits the between and equilibrium of T o e sprubtestliesenergy satellite’s the perturb us Let . .D Pinotsis D. A. r(r/r r o . o ω 1 n period and 1 ) hl anann constant maintaining while , ω o /ω T r 1 E o 1 h aelt will satellite The . = − yatn amount tiny a by T D 1 e /T and = o K = r o Γ (we 1 + / KA D , Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 hs emtia oeswr netdaddmntdteefe eas ftedel rooted deeply the of because thereafter dominated and invented were models geometrical These certain a to coincide which authors, modern by given been have arguments final some Thus, paetobto h u.H nwta h orsaoso h eradepcal h arcs the especially and year the of seasons four the Earth. the that interpret the knew to world, He order in the E Sun. time of the that of centre at because available orbit the circles were apparent eccentric that to observations the respect the of used with theory also symmetrically Hipparchus the move than epicycles, bodies of preferable of theory celestial more the model the was About the [7]). planets later it of and the that 188 circles (pp. for said eccentric him Hipparchus used of to model belonged first the claimed, to he Moon he order the model, which, epicycles, and In mathematical Sun theory. a the for heliocentric using especially by the and system rejected geocentric and [5]), the of theory 115 explain geocentric (pp. work the previous in adopted analysed arguments Hipparchus on based century, following the During 3.2 introduced considering Apollonius of why instead reason epicycles orbits. the elliptic the depict be with bodies must together celestial the This them that explanation implemented planets. satisfactory and the first circles of a bodies eccentric orbits celestial provided the apparent of consequently orbits the the They that of motions. and world circular the of normal centre the were at still was Earth the that belief interpretation. mathematical its towards horizons new created and novel were important circles one eccentric models. of geometrical consisted and motions, ideas circular Earth cosmological normal the of being with evolution of together the still immobility towards world the is step the however, retained of it problem, centre although This the system, [5]). at that of of development 113 lived The who (pp. discussed. astronomers, previous Apollonius or and contemporary Heraclides other some between to excluding course belongs simultaneously of it possible, without that is which Apollonius, possibility It to certainty. Sun, belongs the with system known the this not the of is around authorship system system the circles this that that however, of namely inventor eccentric to The century, According or Earth. fifteenth the times. epicycles around the ancient moves on in in move known developed planets. planets was became can exterior five Brache, that one three Tycho system the historians, of the for system modern to implemented the and similar also writers system was Heraclides a ancient of Hence, certain theory of the that study conclude thorough a After authors. degree. ancient of fragments corresponding the studied and that Schiaparelli, do astronomers to the tried as Tannery such and historians Heath Some was. inventor its who and Greece ancient w oesaottemto fteSun. the of motion the about models two i)Drn h rpclya,teSnHmvso neiyl iharadius a with epicycle an on moves H Sun the year, tropical the During (ii) dtrie arywl h paetmto fteSnwt nerro iueo h arc the of minute 1 of error an with Sun the interpret could of theories motion two apparent these of the Each well circle. fairly eccentric (determine) an along motion the with identical i uigoetoia er h u oe,wti osataglrvlct,aogone along velocity, angular constant a within moves, H Sun the year, tropical one During (i) γ ed o nw oee,frsr hte ope hoyo h lnt a eeoe in developed was planets the of theory complex a whether sure for however, know, not do We ntecneto h ecnrcsse,tegoerclmdl fteeiylsadthe and epicycles the of models geometrical the system, geocentric the of context the In epoe htteetotere r qiaet enn httemto na pcceis epicycle an on motion the that meaning equivalent, are theories two these that proved He , h eteBo h pcce eit ietdrcin(atad)addrn h same the during and (eastwards) direction direct a depicts epicycle, the of B centre The K fiue2). (figure radius with Earth the around circle eidtedfrn icewt radius with circle deferent the period γ h dpino h ecnrcsse yHpacu fRhodes of Hipparchus by system geocentric the of adoption The E Γ ,E = eR γ and rmteobserver the from γ fiue2 eenteul hsieult e i otefruainof formulation the to him led inequality This equal. not were 2) (figure E nin re omlgcliesadmteaia models mathematical and ideas cosmological Greek Ancient Γ fo h at) where Earth), the (from R R = n centre and Aacnr ,wihi lcda distance a at placed is which K, centre a KA Γ h at fiue3). (figure Earth the , e steecnrct fteorbit the of eccentricity the is r = eR westwards. 477 Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 h atranraiyi u otecag ntecraueo h ri n hrfr nthe in therefore and orbit the of curvature the in change the to due is abnormality latter The hsrsl oehrwt te bomlte fteMo’ oini u otetdlforces tidal the to due is motion Moon’s the of abnormalities other with together result This fteecnrccrl oad h ieto fteacwihi on enn ewe the between meaning eccentricity found the is for centre which values the arc right placed the the he of seasons, direction circle. the the equinox eccentric of spring towards an lengths circle was the eccentric Sun between the the differences of of the orbit interpret to the order that In assumption the made correctly Hipparchus ftetomtosfrcranpaes codn oPoey( Ptolemy to equivalence According the prove planets. to certain tried for who motions first two the be the we to of years.As seems 1700 for Apollonius acceptable mentioned, remained it already that have small so was inaccuracy That [13]) of 163 (pp. 478 napro fapoiaey9yas enwko httertto fteln fapsides of line the of rotation rotation the 40 complete is that perigee, a orbit know the performed Moon’s now and of the We of advance east years. plane to the 9 the west that approximately on from knew of shifted also period being He a was in [46]). apsides, of of 134–136 line pp. the [45]; of 3–14 pp. h nlnto,teolqiyo h citc a o 23 not was ecliptic, the of obliquity the inclination, the ete acltdfis bomlt fteMo’ ri wihi rtapoiaini an is approximation to first equal in it (which found and orbit Moon’s eccentricity the its namely of ellipse), abnormality first calculated then He formulated also He Moon. abnormalities. the many of motion includes the it about that models and two motion Sun’s the than complex Kepler. of while laws ecliptic. ecliptic, two the the towards with inclined coincided was Sun orbit the the the angular of that of different region considered orbit with the astronomers the in moving previous faster that was and proved Sun apogee he the the Additionally, towards since perigee). (slower year, circle eccentric of the seasons along four velocities the of inequality the onie ihteelpi.Tepaeo h pccefrsa nl f5 of angle an forms epicycle the of plane The ecliptic. the with coincides hc oe nthe on moves orbit. which its of apsides of line the through Sun eccentricity. the of transitions two between time the to equal qio n ,i h angle the is A, and equinox e rttle bu h pgeadperigee and was apogee the about talked first codn oEaotee’cluain 23 calculations Eratosthenes’ to according ssitdo ohsdso t vrg oiinu oawdho 1 of width a to up Moon position due the average is Hence its which position. average of orbit, its Moon’s sides around the both apsides epicycles. of of on of abnormality line theory shifted the second the is of implement the the motion should determine regressive exactly he the to is that to able This conclusion explain then orbit. the possibly Moon’s was to not led the He could was of it he understood abnormalities because why therefore other inadequate reason He was the Moon. circle meaning the eccentric observations, upon the the acting of planets) theory the the and that Sun the by attractions (the eeetcrl n hrfr nescsteelpi ta nl f5 of angle an at ecliptic the intersects therefore and circle deferent t raetdciaini 23 is declination greatest its acltdta h nlnto fteMo’ ri nrlto oteelpi s5 is additionally ecliptic He the zone. to zodiac relation in within orbit found Moon’s totally the is of inclination and the sphere that celestial calculated the on circle great a ic t urn au is value current its since = sfra h ri fteMo scnend iprhske htismto smore is motion its that knew Hipparchus concerned, is Moon the of orbit the as far As ntal,h osdrdta necnrccrl ol eitteaprn ri fteMoon. the of orbit apparent the depict could circle eccentric an that considered he Initially, first the to equivalent was Hipparchus of theory the approximation, first a as Consequently, iprhstu cetdta h onrttscokieo neiyl,tecnr of centre the epicycle, an on clockwise rotates Moon the that accepted thus Hipparchus iprhs( f[1;p 6 f[3)osre htteobto h onapast be to appears Moon the of orbit the that observed [33]) of 265 p. [41]; of 7 (E Hipparchus 0 e . = 17 eas on httelnt fteaoe,wihi h r ewe h spring the between arc the is which apogee, the of length the that found also He 0167. 1 / 24 = γ 0 n h umrsltc ,maigtedrcino h pgeA n chose and A, apogee the of direction the meaning E, solstice summer the and . 46 hc a eystsyn,sneiscretvlei approximately is value current its since satisfying, very was which 0416, deferent e = (γ ◦ 0 Γ 51 onecokieaon h at.Tepaeo the of plane The Earth. the around counterclockwise . 055. A ) + = e 5 n h egho h pge(.4 f[3) Hipparchus [43]). of 41 (p. apogee the of length the and ◦ 65 ◦ = 40 ◦ .D Pinotsis D. A. 30  28 ◦ 51 35 loavr odvle nta a eexplained he way that In value. good very a also , fteSnsobtadfudta t eccentricity its that found and orbit Sun’s the of ◦ 51 prxmtl i 2 7o 4] .4 f[44]; of 40 p. [41]; of 67 22, (i, approximately e erads thsapro f88 years. 8.85 of period a has it so and year per prxmtl.A h ieo Hipparchus, of time the At approximately. e = ◦ 0 . 27 85 hsi eygo value, good very a is This 0875. ietdy u 23 but today, like , Γ f[1;seas [42]). also see [41]; of 4 , ◦ ec,the Hence, . ◦ 2 n ihaperiod a with and .25 ◦ ihtepaeo the of plane the with ◦ n therefore and nodes ◦ deferent 43 move and i.e. Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 hs bomlte eeepandol fe h neto ftetlsoeado h law the of and telescope the of invention the after only explained were abnormalities These n auncudb ut aiyadacrtl explained. accurately and easily quite be could Saturn line the and of middle the in placed is which eeetcrl was circle deferent and o h ai ftedfrn iceadteeiyl eue h olwn values: following the used he epicycle the and circle deferent the of radii the radius For the of length the calculated rmes ows n efr opeertto rudteai fteelpi naperiod a in ecliptic the of axis the around rotation complete a 18 perform of and west to east from aldtePoeacsse.Altedtiso i ytmaemnindi i well-known his in mentioned are system his of details the was system treatise All This system. planets. the Ptolemaic of of the motions model the called overcome mathematical upon enlarged to the and upon order modified relied he In which Ptolemy model. Hipparchus, theory, cosmological geocentric own the of his imperfections formulated ideas some epicy- he of the circles, theories on the eccentric meaning Based and Hipparchus, system. cles mostly and geocentric Apollonius Heraclides, the of accepted theories and AD) 100–170 (about Ptolemy Claudius 3.3 and shaping the later undertook form Ptolemy ideas. to Claudius Hipparchus’s him of astronomer help perfecting great would The that theory. observations complete accurate sufficiently a Earth disposal the his around at turn would have however, which of centre the epicycle, the the on on move would of planet some each interpret that to order in formulated remarkable. he are that provided orbit theory he its the that of and results abnormalities Moon and the the calculations of the however, motion this, the all about Despite Newton. satellite. by our gravity of of motion the of abnormalities the all interpret or determine angular possibly not these could in he difference the consequence, the month. of nodical a motion or As direct draconitic years. the the of 9 justifies because approximately velocities coincide in completely not apsides did of [32]) line of 243 and 240 (pp. day o rmteErh u ahrfo h hr on ,the E, point third the measured from be rather should placed but circle deferent Earth, the and the on moves Jupiter from epicycle Mars, not each of Sun, centre angular Venus, (uniform the Mercury, velocity which angular to by constant motion) Moon the would that the considered circles Ptolemy from eccentric 5).Also, goes The (figure Saturn one circle. eccentric as an radius therefore circle their Earth was deferent increase the circle own with deferent its Each coincide had Earth. not planet the would Each of which centre circle. K, the deferent epicycle, centre the an with on on eastwards moving moving therefore was was It which motions. circular the normal predicted two taneously and interpreted planets. and system Sun This the Moon, circles. of eccentric positions apparent and motions, epicycles circular of normal combination of superposition the Greek upon ancient the of world. knowledge western of the spreading and Arabs the the to to contributed astronomers treatise This volumes. 13 of sfra h td ftemto ftefiekonpaesi ocre,Hpacu assumed Hipparchus concerned, is planets known five the of motion the of study the as far As codn ota ytm ahpae a oaigaon h at ypromn simul- performing by Earth the around rotating was planet each system, that to According oehrwt h rvosmteaia-omlgclmdl,tePoeacsse relied system Ptolemaic the models, mathematical-cosmological previous the with Together eas ftefwadsml srnmclisrmnsta iprhshda i disposal his at had Hipparchus that instruments astronomical simple and few the of Because r ( luisPtolemy Claudius = 2 deferent tteohredo h diameter the of end other the at / Syntaxis 3 5 ) ( 1 = / 4 18 ) nti a,Hpacu adtefudtoso lntr hoy eddnot, did He theory. planetary of foundations the laid Hipparchus way, this In . epciey therefore respectively; , . or er.Ti bomlt scle ‘ called is abnormality This years. 6 Almagest ω 1 nin re omlgcliesadmteaia models mathematical and ideas cosmological Greek Ancient = 13 ◦ ( eaeSnai e Astronomias tes Syntaxis Megale 10 35 r fteMo’ pcceadtepsto fteapogee. the of position the and epicycle Moon’s the of e a n hto h epicycle the of that and day per Γ e onn h at otecnr fthe of centre the to Earth the joining .Teeoe h ergaemtoso as Jupiter Mars, of motions retrograde the Therefore, E. = 5 ( 1 / 4 ) : 60 h ersino h nodes the of regression the i.e. = Γ siltos hc rs rmthe from arose which oscillations, 0 qatpoint equant ic twudb lcdoutside placed be would it since , . 85 h nua eoiyo the of velocity angular The 0875. e 4] 4],wihconsists which [42]), [41], see hspitsol be should point This . ω 2 = eeetcircles deferent 13 .Hipparchus ’. ◦ 3 R 54 = per 479 60 , Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 h ai fthe of radii The iue5 h omlgclmdlo tlm.Ki h eteo the of centre the is K Ptolemy. of Earth. model the cosmological of The centre the 5. Figure 480 h at,bcueteepaesoclaefo n ieo h u,t h te.Tegreatest The other. the to Sun, the of 28 side exceeds never one Mercury from of oscillate elongation planets these because Earth, the fe h neto ftetlsoe ae,hvn ayosrain thsdsoa,Kepler disposal, his at observations many 1610 in having system Later, heliocentric telescope. the the of of proof invention first the the only achieved after the Galileo remained system. and was foundations heliocentric prevailed the system the laid it Copernicus Ptolemaic of why until Ptolemy, The reason after years the position. 1400 is for apparent system that planetary its acceptable and to system well of flexible resultant centre fairly very the the a fit that therefore way of could a and planet such epicycle in the combined the of and on modified motion planet be could the circle of deferent the velocities planets. on angular epicycle the the of circle, positions deferent apparent the future to the with coincided that statements stars. of the his of rotation that sets so The and system planets. rises his the all initiating of was circles h 24 deferent in the Earth the surround around which motion. sphere daily sphere, their a in in participated they riveted time and same Moon the the At of direct. motions always the contrast are in because Sun considered, needed the Ptolemy not Sun. of the were for epicycles year that a ideas, and Hipparchus’ Moon the with for month a of period a within the on Pontus. based of system Heraclides this of invented theory Ptolemy the that of conjecture existence way, the one either make optimal disc, on therefore an solar appears may the planet in We other. each of described velocities result, side be a angular other As could with sphere. the of positions celestial epicycles, or the centre apparent their on the their projected on are on that rotate orbits always way their planets projected a two are such the planets in as two arranged follows: secondly, the as and, of obtained Sun epicycles be the the can of conclusion centres This Venus. the and firstly, Mercury about Heraclides of circle. deferent theory corresponding the the connects around which move epicycles line the corre- straight of radii same centres the the different as along of Sun, found the epicycles always to the Earth are of planets centres two the the to that sponding considered Ptolemy reason, this For sfra h neirpaesaecnend hynvrsa a rmteSna enfrom seen as Sun the from far stay never they concerned, are planets interior the as far As h onadteSnwr oigaon h at nteronecnrccrl,eastwards circle, eccentric own their on Earth the around moving were Sun the and Moon The the in contained that to equivalent then is perform planets two the that motion resultant The sn h bevtosta a encletdutlhstm,Poeycudawy improve always could Ptolemy time, his until collected been be had that to observations found the Using are stars the that supposed Ptolemy stars, fixed the of motions the Regarding deferent n the and epicycle i.e. hyse ooclaefo n ieo h u othe to Sun the of side one from oscillate to seem they ◦ .D Pinotsis D. A. h retto ftepaeo h pccerelative epicycle the of plane the of orientation the , n hto eu ee xed 48 exceeds never Venus of that and eeetcircles deferent ,Eisthe ◦ qa point equal approximately. and Γ is Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 h s fmteaisad npriua,goer e osgicn otiuin towards contributions significant to led geometry particular, in and, mathematics of use The at rudisai.Smtigsmlri lomnindi rsol’ book Aristotle’s in mentioned also is similar Something axis. its around oscillations. hannonic Earth simple of superposition the circular normal by of words, superposition other the believed by in Plato described or, be Simplicius, motions could to planets According of time. motions Plato’s apparent as the that early as models cosmological of tion used be could philosophers belong scientists. and modern they scientists by since ancient basis scientists, other compare philosophical Greek or to a Plato ancient try as of or to ideas Plato inappropriate The of be eras. would different those to it with Bang that Bondi theories note Big Gold, and should of the ideas We theories Dirac. with modern cosmology of the accordance that with in consistent and are not Hoyle are and and in they relativity hand, Einstein other of by the introduced theory On ideas general model. cosmological the same modern the of to at vanish formulation close will are the they Plato that and of matter views of These existence time. the with connection in overcome.’ ‘sky’) could it [geocentric] model named model new This a that time. point the weak at any ignored theory not have was be dominant to Aristarchus may not the a of ideas seemed was at model new model appears heliocentric then geocentric problem The field, the a forgotten. science to because relevant be solution the and the in known Kuhn dispute ‘If widely this. much others: become to not among contributed is noted cycles there [47]) when eccentric of time of 148 and and 141 epicycles 140, of (p. models by the supported well Also, was as Ptolemy. it Hipparchus as because Eudoxus,Apollonius, was Plato,Aristotle, This Pythagoras, of elsewhere. intellects presented eminent be the will requires which fact this analysis, of detailed explanation a Copernicus.An of time the until model the cosmological dominant mentioned Philolaus when that, assume we order In Simplicius, Fire system. of Central heliocentric consideration the this introduced who explain hand, Philolaus other to was the it Hicetas, On that thinking. Philolaus, Aristarhus’ considered intellectual influenced Anaximander, Simplicius fervent who namely and Pontus, not Aristarchus, of long Heraclides of did a and of precursors theory in Ecphantus product the heliocentric only the by the was not However, but created horizons model general. activity new mathematical in created peculiar a thought that as human idea arise innovative in an also was but Samos astronomy of Aristarchus of celestial esis that considering of the instead be orbits. epicycles could elliptic with This in travel together motions. bodies eccentric them circular introduced sections, implemented normal conic on and were work world bodies circles definitive the the of celestial wrote centre who the the Apollonius, at why of still reason orbits was Earth dominated the the and that that invented belief and were rooted deeply models the geometrical of These because inter-thereafter system. mathematical geocentric the in the horizons of eccentric new of created pretation and and epicycles models of geometrical models complex were The theories. circles and ideas cosmological of development the Conclusions 4. laws three Finally, the Sun. theoretically proved the 1686, around in gravity planets of the Kepler. law of of the motion of discovery of the laws after three Newton, the formulated and discovered hruhsuyo nin ore n oenatosipista h eicnrchypoth- heliocentric the that implies authors modern and sources ancient of study thorough A ehv ocue htPaoblee nasmlaeu einn ftm n pc (he space and time of beginning simultaneous a in believed Plato that concluded have We h rnil fidpnec n uepsto fmtoswsitoue nteformula- the in introduced was motions of superposition and independence of principle The ae nafamn of fragment a on Based the became finally system geocentric the hypothesis Aristachus’ of importance the Despite eipidteSun. the implied he , nin re omlgcliesadmteaia models mathematical and ideas cosmological Greek Ancient Timaeus ecnld htPaoacpe h al oino the of motion daily the accepted Plato that conclude we nHeaven On 481 . Downloaded By: [Bochkarev, N.] At: 16:09 7 December 2007 1]RD Hicks, R.D. [10] References an considerations. to equivalent similar is make This astronomers Today, stars. Universe. the the by planetary of occupied the size space and the the Earth of with the estimation comparison of actual orbit in the the point conceived by a occupied Aristarchus as space that system the conclude considered we He analogy Universe. of this size From sphere. celestial the of Kepler. of laws two energy satellite’s the Earth constant. perturb the momentum round we angular goes if the which orbit, satellite maintaining a circular of stable motion the a in in arises circle the eccentric describing an to system conditions, his formulate to order in Venus. Pontus and of Mercury Heraclides of motions of theory were ideas the Ptolemy’s of that on motion conjecture theory the therefore based to may the We equivalent Heraclides. is to of planets theory According two the these epicycles. from of resulting of each by theory executed the motion that resultant of the oscillations Ptolemy, inventor harmonic the two therefore was of Heraclides superposition a is planet eight. each of of figure motions a spheres the resembles from last resulting two motion the the example, For of motions. of superposition of principle the students. his to phenomena. celestial need explain this to emphasized order also in models He mathematical of formulation the in especially world. of chapter different a In 482 1]Plutarchi, [11] [17] Plutarchi, [16] Dicks, D.R. [14] Dreyer, J.L.E. [13] in Pinotsis, A.D. [12] 1]Aristotle, [15] 9 ignsLaertius, Diogenes [9] Heath, T.L. [8] Smyrna, of Theon [7] in Pinotsis, A.D. [6] Pinotsis, A.D. [5] Lloyd, G.E.R [4] Hermann, G. [3] Kern, O. [2] 1 .Chassapis, K. [1] rsacu omltda nlg ewe h iesoso h lntr ytmadthose and system planetary the of dimensions the between analogy an formulated Aristarchus initial certain under equivalence, its and motion epicyclical the of analogue mechanical A realized also had Eudoxus that conclude we spheres homocentric of theory the studying After iprhs hoyo ceti ice s safis prxmto,euvln otefirst the to equivalent approximation, first a as is, circles eccentric of theory Hipparchus’ Venus. and Mercury of motions apparent the explain to first the was Pontus of Heraclides and astronomy in mathematics use to need the foresee could Plato that noted Simplicius 1964). 1981). 279–307. pp. 2003), Athens, Exarchacos, (Th. 2003 October 97–137. pp. 1998), Circles Rhodes, Eccentric the and Epicycles of 1991). Cambridge, 1967). ltrh Maralia Plutarchi 1895). Leipzig, 241–246. pp. 2001), Samos, Seimenis, (J. 1893). Leipzig, Bernardakis, N. (Gr. rhcrmFragmenta Orphycorum nHeaven On ltnceQuaestiones Platonicae ePaii Philosophorum Placitis De ie fEietPhilosophers Eminent of Lives rsacu fSamos of Aristarchus al re srnm oAristotle to Astronomy Greek Early Orphica itr fAtooyfo hlst Kepler to Thales from Astronomy of History A h esn hc e iprhso hdst dp h ecnrcSse n h Models the and System Geocentric the Adopt to Rhodes of Hipparchus Led Which Reasons The h rhc,teGekAtooyo h ’Mlenu B.C. Millennium B’ the of Astronomy Greek the Orphics, The ehd n rbesi re Science Greek in Problems and Methods rceig fte4hHlei ofrneo rsacu fSamos of Aristarchus on Conference Hellenic 4th the of Proceedings rceig fte4hAtooia Conference Astronomical 4th the of Proceedings eie n orce yC uet(ebe,Lizg 1954). Leipzig, (Teubner, Hubert C. by corrected and revised , hlspiPlatonici Philosophi ia Philosophorum Vitae ( SmtbsCsaiFish epi,1805). Leipzig, Fritsch, Caspari (Svmptibvs ερ Timaeus ıO ´ νρανo ´ ( Wimn,Bri,1922). Berlin, (Weidmann, λατωνıκ Ofr nvriyPes xod 93 DvrPbiain,NwYork, New Publications, (Dover 1913) Oxford, Press, University (Oxford lt em ocnie h at stefie eteo the of centre fixed the as Earth the consider to seems Plato , ν ¯ oeaeinCrnce,vlm E(soito fClueadArts, and Culture of (Association IE volume Chronicles, Dodecanesian , ,tasae yP aaitu(e yoa tes 1989). Athens, Synora, (Nea Panagiotou P. by translated ), eie yE.Hle Tunr epi,1878). Leipzig, (Teubner, Hiller Ed. by revised , ( ( B ερ o ϕıλoσ ıoı HradUiest rs,Cmrde ascuet,1925–1938). Massachusetts, Cambridge, Press, University (Harvard ´ .D Pinotsis D. A. ı ´ αZητ ´ Gektasain Tae n usn odn 1970). London, Hudson, and (Thames translation) (Greek ρσκ Aρεσ ηµατα ´ oϕων ´ νω ıλoσ oντων ´ Gektasain CmrdeUiest Press, University (Cambridge translation) (Greek n dto DvrPbiain,Nwok 1953). NewYork, Publications, (Dover edition 2nd , ,tasae yHS og(lrno rs,Oxford, Press, (Clarendon Long H.S. by translated ), ,Piah oai,vlmV G.N Bernardakis, N. 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Hist. Int. Archs itr fAcetMteaia Astronomy Mathematical Ancient of History A opsto Mathematique Composition ytxsMathematica Syntaxis nin re omlgcliesadmteaia models mathematical and ideas cosmological Greek Ancient dtdb .Emndbr HradUiest rs,Cmrde ascuet,1933, Massachusetts, Cambridge, Press, University (Harvard Edmunsdsbury S. by edited , ( Mετ ( αµµ . Ensu ercsGogu epl,Lizg 1789). Leipzig, Leopold, Georgius Henricus (Ernestus . ανσıκ ´ ıτης ´ o.I enmn,Lno;HradUiest rs,Cambridge, Press, University Harvard London; Heinemann, I, Vol. , oueI(re rnlto)(lioHel,Mln,1971). Milano, Hoepli, (Ulrico translation) (Greek I volume , 33 ,vlm ,eie yE tmts(ehia hme fGec,Athens, Greece, of Chamber (Technical Stamatis E. by edited B, volume ), α ´ (1983). 3 Kko,Ahn,1992). Athens, (Kaktos, ) rnltdb ..Hieg(ebe,Lizg 88 1900). 1898, Leipzig, (Teubner, Heiberg J.L. by translated , rnltdb ..Hla(hH rn,Prs 1813). Paris, Grand, (Ch.H. 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