Anaxagoras, Eudoxus and the Regression of the Lunar Nodes

Libre JHA ii (1971), 23-28 Aa N vie: SAYS ANAXAGORAS, EUDOXUS, AND THE REGRESSION OF THE LUNAR NODES* VICTOR E. THOREN, Indiana University In the century and a half since Ideler? first advanced his claims. Eudoxus has come to be accorded a prominent place in the history of astronomy. Although very little is actually known regarding the details of his work, the hippopede is notorious to historians of science. and no one now doubts that Eudoxus took the first serious step in the development of planetary theory. Exactly how much he accomplished, particularly in the task of establishing accurate parameters, will probably never be Known. But that he must have done a fairly rigorous job was documented beyond a reasonable doubt nearly a century ago. through the ingenious re-constructions of Schiaparelli.? Given only the numbers of spheres involved, and vague explanations of their arrangements and purposes, he showed how the various systems could have worked. in what respects they would have failed. and how these inadequucies could have been met through subsequent modifications Callippus. known to have been intreduced bs Schiaparelli’s work was almost completely extrapolative: he started trom the reports of Aristode and Simplicius.* then went beyond them to develop geometrical combinations and numerical constants (which latter, of course. were Wholly idealized. and never taken seriousls by him or anvone else). one respect. however. Schiaparelli felt justified in doing more in In actually altering the information at hand. The situation arose in the lunar theory." which was represented by three spheres. The outermost, as in all of Eudoxus’s other theories, provided the diurnal rotation, and posed no problem. But according to both the text of Aristotle and the gloss of Simplicius. the second or middle sphere had its equator in the ecliptic, and moved with the monthly (draconitic, ideally) motion: while the innermost sphere. carrying the Moon on its equator. was slightly inclined (3°, hopefully) to the second. and slowly, moved retrograde very Now, neither Schiaparelli nor Ideler before him had the slightest doubt as to what Eudoxus was trying to do. confirmed Indeed, Simplicius had expressly their instinctive conclusions by stating that the third sphere was necessary to account for the Moon’s movement in latitude, and for the fact that the Moon does not always reach its greatest latitudes at the same zodiacal points. but in places which shift steadily westwards. Unfortunately, the latitudinal motion produced by the reported arrangement is most peculiar compared to the actual behaviour of the Moon. through its latitudinal cycle each For instead of propelling the Moon month, what the purported mechanism would do is carry the Moon through its monthly longitudinal circuit at a nearly constant latitude(!) and keep it alternatively north and south of the ecliptic for half the period of the slowly rotating (18-6 years?) inner sphere. of the blatant contradiction with * the phenomena, tensred during the the tenure of National Science Foundation Grant GS 2928, Because Ideler and Schiaparelli

Journal for the History of Astronomy The Regression of the Lunar Nodes reversed the actions of the middle and inner spheres, so that the middle one moved slowly retrograde and the inner one provide d the monthly motion. For as long as historians of astronomy have been reading accounts of Eudoxuss system, this reversal has seemed both reasonable and justified. Only recently has the first challenge been issued. by D. R. Dicks. in his Early Greek astronomy to Aristotle’ As part of a general and laudable attitud e of scepticism toward the interpretations that have been placed on various features of early astronomy by enthusiastic commentators, both ancient and modern, he severely criticizes this past willingness to “unhesitatingl y take Simplicius’ explanation of the third sphere as referring to the regression of the lunar nodes”. Characterizing this assumption as “yet another exampl e of a misleading interpretation of early astronomical thought by attributing to it concepts which were only discovered much later”, he regards it as “highly improbable that Eudoxus knew anything about the nodes... ”. This difference of opinion constitutes the subject of this paper. So far, all that exists is a difference of opinion. If the technicians never bothered to present an explicit argument in favour of their conclusions, neither has Dicks offered any counter-areument in favour of his. At the risk of being guilty of piling specula tion on speculation I wish to defend Ideler’s emendation by analyzing the situation bevond what has thus far been done. | o | If there is anything that can truly be called a datum | in pre-Socratic astronomy. it must surely be that Anaxagoras provide d. if not for the first ume. then at least to the first general satisfaction. an explana tion of the phases of the Moon." Closely related to this tradition is the report that he also explained the cause of eclipses.” Now, while the authority for this part of the story is considerably less Impressive, there are at least two reasons for taking it seriously. The first is the fact that the reporter (Hippolytusy include d in hisaccount an elaboration on the theme (to wit. that lunar eclipses are sometim es caused bs bodies other than the Earth)" which renders it considerably less likely that he was just carelessly endowing an ancient figure with recent ideas. The second is the logical relationship between phases and eclipses. With the crux ef both beine the shining of the Moon by reflected sunlight—somet hing explicitly credited 10 Anaxagoras by Plato.” whose knowledge would have been practically first hand— It is surely not stretching the imagination unduly to believe that Anaxagoras might actually have followed the consequ of eclipses. ences through to the explanation Unfortunately. of course. the parallel is rather too complete. For while the phases cycle around every month, nobody needs to be told that eclipses occur a great deal less frequently. The fact that some would take place during daylight hours may have occurred to someon e: we have strong reasons for believing that the Babylonians had already recognized this difficulty." But there would still be a lot of phantom eclipses involved in this so-called explanation. To avoid the ridicule of his fellow philoso phers (and the satire of impertinent dramatists) Anaxagoras would have had to eliminate them. That he actually did so is suggested by Hippolytus's report of “other bodies": for it is hard to imagine Anaxagoras working to get extra eclipses if he already had them occurring every month"? It would be nice to be able to assume that he got rid of the superfluous eclipses by inclining the orbit of the Moon to the ecliptic, but that happy solution seems untenable. 25 For, if it did not still leave eclipses occurring more frequently than the Greeks noticed them (they must have missed at least a few) a complete grasp of the situation would certainly have obviated the necessity of introducing sub-lunary bodies to generate extra lunar eclipses. If we are to take these bodies seriously, We must assume that they were invoked to account for the fact that Junar eclipses are seen so much more often than solar eclipses. All of this would indicate that Anaxagoras probably (and not surprisingly) lacked an appreciation of the immense size of the Earth relative to the Moon’s shadow, and had to restrict the frequency of his primary occultations to something commensurate with the frequency of solar eclipses. Under such circumstances, there would have been little else to do but call upon other bodies to provide the required lunar eclipses. lt would appear, then, that Anaxagoras probably failed to work out the entire potential of his theory. The alternative is to assume that the Greeks had not at that timesucceeded in distinguishing astronomical from meteorological phenomena. and that the “other bodies” were hypothesized to contend with occasional cloud-coverings. It is one thing, however. to suppose that Anaxagoras might have contented himself with some vague postulate whereby true eclipses happened only at occasional new and full Moons. and quite another lo assume that such an incomplete explanation would stand unchallenged and unextended for any great length of time. Sooner or later. someone would have to reach the concept of an inclined orbit. Dicks (p. 179) is willing to believe that it might have arisen empirically, through observation of the Moon’s path among the stars. Whether it is as easy as Dicks assumes te “observe” the ecliptic and various deviations therefrom. the fact remains that once the inclined orbit is attained. the concept of the node is there, Such a concept would provide a powerful generalization of eclipse phenomena: viz.. eclipses would perpetually occur at two opposite seasons of the year. in two opposite points of the heavens. Experience. of course, would soon falsify this proposition: the premisses would have to be modified to account for the fact that the eclipse positions move slowly backward through the vear and around the zodiac. Voila! the regression of the lunar nodes. Now, how reasonable is it to suppose that the Greeks might actually have followed some such train of thought? What kinds of problems are involved in assuming that, by Eudoxus’s time at the latest, they had managed. through eclipse considerations, to formulate the concept not only of the nodes themselves. but of the regression of the nodes as well? One difficulty, obviously, is the lack of any real proof that pre-Socratic philosophy (or astronomy) took eclipses seriously enough to invest a certain minimum amount of time and effort in noting and remembering their occurrence, and worrying about explaining them. To this point, one can at least reply that eclipses appear to be second only to things such as earthquakes in their capacity to impress primitive man. and that we know from Herodotus's report about Thales that the Greeks had already accepted the ideal of predicting their appearance well before Anaxagoras's day." Beyond this more or less a priori point stand Dicks’s two objections. The first is the (as he sees it) abstract nature of either the concept of the node, or the conception of its movement. This attitude is, to say the least, puzzling. Dicks is willing to concede the notion of an inclined orbit. How, then. does

ioni of their element on the subject In questT generations of people (equally out ate D a an in nted prese m anis mech read a brief descripuon of an intricit (as, indeed, Simplicius cran Sir a without finding any problem in s of succeeding technicians wer pal del or Eudoxus blundered and generationreali zing. or at least taking him tc +IE his system into better ones without a really egregious error. The SOT ISSchiamen for. the fact that he had committedpuzzl pi e lies in following Ideler and either way. But surely the lesser one have an inclined orbit without having nodes? and how much of a geometer does one have to be (Dicks labels Eudoxus a mathematician of genius) to recognize that the one entails the other? As for the regression of the nodes. Dicks again cheerfully admits that Eudoxus's theory was set up “to explain why the maximum deviations north and south did not occur always at the same points of the zodiac”. Yet, despite this reference to the movement of the limits. and Simplicius’s expressed statement that the limits “shift steadily westwards” (retrograde), Dicks balks at assuming a regression of the nodes. 27 The Regression of the Lunar Nodes Journal for the History of Astronomy in their reversal of the periods of the two spheres. Given the indissoluble association between the nodes and the limits, the only reasonable REFERENCES way to deny the regression of the nodes is to reject Simplicius’s comment on the westward shift of the limits. 88. IR: o 49on 33 1830, 212; 189! ” 98, Li \ 1825, Hi phil. sie er Akademie,ie, Hist.I. Ludwig Ideler. Abhandlungen der Berlin Si se Dur in. Brera di io vator ne del Reale Osser arelli. Publicazio i a ppus >. Giovanni Schiap tr Pe des ren Sphac schen entri homoc “Die Horn. W. by man trans. Yet this is precisely what Dicks is unwilling to do—tamper with Simplicius's statement. . u "Ger . y INT und des Aristoteles”. Abhandlungen zur Geschichte der Mathematik | ON o 3. In his Meraphysics (A. 8), Aristotle say ake po ın mn the Sun and Moon Eudoxus held that the motiotn isofthat Bee second revolves MURS€ ThE> seco sturs. the of theeffixed shite, In this last issue, of course, we encounter the second and more substantial of Dicks’s objection to Ideler’s reversal of the second and third spheres. Now. one can scarcely argue with the proposition that we must make historical as well as scientific sense of the data available to us: obviously. re-writing ancient reports to conform with our ideas of propriety is a serious step. ormas mo hich the outer 5 of| ol w whic spheres, ere? bs LA à the third revolves in a circle which jac.. and bise s the: zodiac ‘hich bisect which i circle Mmoves iats incited al i Ente Moun the which in role ci the but c: lac: e zodia the breadth of the motion LEvi DE came that the ee the Sun moves, And he held def w n which i that in s angle than first(Gre and an seve nd ii he sa But so is rejecting ancient reports. and Dicks is quite properly willing to do that when the occasion seems to demand it. Elia s: 4and that of these the snheres: as four sphere 1 each case involv inv ed in is that which ehh ETS sl Pin ner e of the fixed stars has its> matio iù as before (for the spher c es IE virele the in R n i m which order. ¢ n i In i next eree Here would seem to be such an occasion, Consider the options. On the one hand is the possibility that Aristotle's account is correct. and that Eudoxus just blundered. It certainly cannot be A thee spher r ». and sphere on to all the planets): the € ac. isIS comm > zodiac (Loe thi ined to the equator off thethe third. ef] eu o . ugh genera iban 6.o See Heath. 26 7. Altho , usual credence. pi the later reports of his ideas can be given N morei than wal Ya e a to be> correc je DIare‚> more. likely which s bution “attri his g 7. Dicks tp. 591 includes this amon Following included this As nearly as | can sec. Hippolstus (230to A.D.) x, Refutation of all heresies, 1. & sake constitute a targel of completeness. tl does not scem now’. moved by four spheres in each case: the first and second of these are the same as for the Sun and Moon... .0.!* Note Aristotle's generalization. For the planets. (to say nothing of Aristotle) ever be a comfortable situation. Yet, the fact remains that there was a slip of some kind. Either Aristotle blundered and 498 506 fromepp.ed » Drvsene ue ts based isav:on pp.as“178-8 on which this. critiq Press. 1970. The discussion & Cornell University = I t that' in Anaxagoras $ etase, 56) (p. feels Dicks = un lly sceptical. his run-down of the solar and lunar spheres, Aristotle says "The planets are can challenging the direct. neutral statement of a person as alert as Simplicius has contributed to the general before Hipparchus. up and corrected it.!° The alternative. ofcourse, is that the error was Aristotle's Admittedly, this supposition is not without its problems. It is hard to imagine such a slip passing from Aristotle down te Simplicius. through hundreds of vears of readers and commentators. without being noticed by anyone. Nor rding tk small pertions ofit. —that he either misunderstood Eudoxus' system or committed an elementari leaped to an invalid generalization in his own thinking on the subject? \ cp L. Heiberg’s Simplicius: De creto. re situation. which an analogous. bul mush more obscu Actually. the solar theory presentsconfu sion over the possibility th al precession was known : the proposition at hand is not only conceiving that Eudoxus could perpetrate the first sphere conveys the diurnal motion: likewise for the Sun and Moon. For the planets, the second sphere gives the major motion proper to the planet, while the third and fourth spheres together (he goes on to relate) handle the esoteric phenomena: for the Sun and Moon, it should have been the other war around—with the second providing special minor effects and the third providing the main peculiar motion. Is it not possible that Aristotle. the grand synthesizer, here on unfamiliar ground of less than vital interest to him, may have n planets Rn runs tacce entary on this passage runs ‘yecommentary Simplicius's T. L. Heath (Greed asprin. something of this nature, but also that none of his successors ever picked it slip in his account of it. Perhaps it was a combination of these two. a third . sphere 0) son meee moves in the \ circle and the fourth sales in the circle which bisects the zodiaohc. Libra ry\ trans., p. 15%) . rejected a priori: but neither can it be euphemized as “inaccuracy” in the way Dicks does. A system which entails virtually constant lunar latitudes for years at a time.” with the consequence of bunching eclipses during two relatively Jong nodal periods and then ruling them out completely for vears in between is not simply erroneous: it js atrocious. Yet what is involved in information simply for the odie | rhe nai sometintes alsets ofinterp n of the Earth, and gh the interpositio is eclipsed throu os y. “The Moon Moon the when Moon. new the the Moon. The Sun is eclipsed at for polemics in his main discussion, | Io. below (Heath, 26: my italics). Cratylus, 409 A. 3 of Il, See Pannekock’s discussion of the evidence on pp. dI | A history of astronemy (London. ue eclipses. 12. For the sake of simplicity, I shall omit consideration of solar nians. Babylo earlier the with e. 1? As was the case. for exampl ess feat. the tale itself demon: the kernel of truth in the tale of atThal 14. That is. no matter how small concer in Herodotus day, if not least tion. n for eclipse predic 1961). RSeasi;! NA ... neo-n strates the existence of a 1°. o uee atitud sme“ = of the: . moon diction between thisi arrang- ement . n v» deR antl. : iaining to note the contra It is entert have could he that doubt to reason no is "there . Dicks's concession that and give him a me en deviation in latitude. . Land this would mately the Moon's maximumwould the ton Bere which in m syste a invent to like be it of the third sphere”. What ver le serve whate pt attem ans made had one I years. ) ns maximum latitudes every ninel? «Mi È ,9 already in Thales’s. Li h : Ah | a Moan’ s latitudes over a short period of ume. the 4

Journal for the History of Astronomy Nesther Callippus, who generally amehorated the system and specifically added two spheres to the theory of the Moon, nor Eudemus. who is quoted by Simplicius as an independent authority, would appear to have commented on the situation. The later Alexandrians are equally silent. Yet Simplicius (Heath, 68-70) records criticisms of the concentric feature of Eudoxus's system dating virtually from the ume of Aristotle. while Pioiemy does nat hesitate to criticize even the venerable Hipparchus when he feels it ts justified. 17. This translation of Heath (p. 65) expresses the generalization a bit more explicithy than that given in ref. 3. 18. lt would be interesting te know whether Simplicius's detailed agreement with Aristotle is due to his merely following Aristotte and elaborating thoughtlessly on the third sphere, without bothering to check his sources. or whether he may actually have been copying or paraphrasing existing independent documents. Perhaps a Greek scholur could estimate the degree to which this portion of Simplicius’s commentary differs from extemporanevus comments he makes elsewhere. eun 75 i THOREN V, E., Anaxagoras, Eudorus and the regression of the lunar nodes : ef. Aristoteles. | Tuonen V. E., Anaxagoras, Zudoxus and the regression of the lunar nodes : Journ, for the Hist. of Astronomy II (London MacDonald) 1971 23-28. | The representation of Eudoxus’ theory of the movements of the moon, in Aristotle's Metaphys. x11,8, which is commented upon by Simplicius, should he understood as mistaken, contrary to the view of D. R. Dicks (cf. APh XLI 740). This must be postulated in order to reconstruct Greek lunar theory after Anaxagoras’ theory of the phases of the moon. “four net fe; Fl, If; yf Sh ren. ”)