Anaxagoras,Eudoxus,and the Regression of the Lunar Nodes

Ludwig Ideler , Abhandlungen der Berliner Akademie , Hist.-phil. Kl., 1828, 189–212; 1830, 49–88.

Giovanni Schiaparelli , Publicazione del Reale Osservatorio di Brera in Milano , ix (1875); German trans. by Horn W. , “Die homocentrischen Sphaeren des Eudoxus, des Kallippus und des Aristoteles”, Abhandlungen zur Geschichte der Mathematik , i (1877), 101–98.

In his Metaphysics (A, 8), Aristotle says:

Eudoxus held that the motion of the Sun and Moon involves in either case three spheres, of which the outermost is that of the fixed stars, the second revolves in the circle which bisects the zodiac, and the third revolves in a circle which is inclined across the breadth of the zodiac; but the circle in which the Moon moves is inclined at a greater angle than that in which the Sun moves. And he held that the motion of the planets involved in each case four spheres; and that of these the first and second are the same as before (for the sphere of the fixed stars is that which carries round all the other spheres, and the sphere next in order, which has its motion in the circle which bisects the zodiac, is common to all the planets); the third sphere of all the planets has its poles in the circle which bisects the zodiac; and the fourth sphere moves in the circle inclined to the equator of the third. (Loeb Library trans., p. 157).

Simplicius's commentary on this passage runs (according to Dreyer) from pp. 493–506 of Heiberg's J. L. Simplicius: De caech. Heath T. L. ( Greek astronomy , 67–70) has translated small portions of it.

Actually, the solar theory presents an analogous, but much more obscure situation, which has contributed to the general confusion over the possibility that precession was known before Hipparchus.

Cornell University Press, 1970. The discussion on which this critique is based is on pp. 178–81.

See Heath, 26–7. Although generally sceptical, Dicks feels (p. 56) that in Anaxagoras's case, the later reports of his ideas can be given more than usual credence.

Dicks (p. 59) includes this among his “attributions which are more likely to be correct than not”.

Refutation of all heresies , i, 8. As nearly as I can see, Hippolytus (230 A.D.) included this information simply for the sake of completeness: It does not seem to constitute a target for polemics in his main discussion.

“The Moon is eclipsed through the interposition of the Earth, and sometimes also of the bodies below the Moon. The Sun is eclipsed at the new Moon, when the Moon is interposed” (Heath, 26; my italics).

Cratylus, 409 A.

See Pannekoek's discussion of the evidence on pp. 41–7 of A history of astronomy (London, 1961).

For the sake of simplicity, I shall omit consideration of solar eclipses.

As was the case, for example, with the earlier Babylonians.

That is, no matter how small the kernel of truth in the tale of Thales's feat, the tale itself demonstrates the existence of a concern for eclipse prediction, at least in Herodotus's day, if not already in Thales's.

It is entertaining to note the contradiction between this arrangement of the motion in latitude and Dicks's concession that “there is no reason to doubt that he could have noted approximately the Moon's maximum deviation in latitude…and this would give him the inclination of the third sphere”. What would it be like to invent a system in which the Moon reached its maximum latitudes every nine(?) years, if one had made any attempt whatever to observe the Moon's latitudes over a short period of time?.

Neither Callippus, who generally ameliorated 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 time of Aristotle, while Ptolemy does not hesitate to criticize even the venerable Hipparchus when he feels it is justified.

This translation of Heath (p. 65) expresses the generalization a bit more explicitly than that given in ref. 3.

It would be interesting to know whether Simplicius's detailed agreement with Aristotle is due to his merely following Aristotle 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 scholar could estimate the degree to which this portion of Simplicius's commentary differs from extemporaneous comments he makes elsewhere.