17-32. On the general nature of Eudoxus’ theory I cannot do better than quote Heath (p. 195). ‘Eudoxus adopted the view which prevailed from the earliest times to the time of Kepler, that circular motion was sufficient to account for the movements of all the heavenly bodies. With Eudoxus this circular motion took the form of the
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revolution of different spheres, each of which moves about a diameter as axis. All the spheres were concentric, the common centre being the centre of the earth; hence the name of ‘“ homocentric spheres” used in later times to describe the system. The spheres were of different sizes, one inside the other. Each planet was fixed at.a point in the equator of the sphere which carried it, the sphere _revolving at uniform speed about the diameter joining the correspond- ing poles; that is, the planet revolved uniformly in a great circle of the sphere perpendicular to the axis of rotation. But one such circular motion was not enough ; in order to explain the changes in the speed of the planets’ motion, their stations and retrogradations, as well as their deviations in latitude, Eudoxus had to assume a number of such circular motions working on each planet, and producing by their com- bination that single apparently irregular motion which can be deduced from mere observation. He accordingly held that the poles of the sphere which carries the planet are not fixed, but themselves move on a greater sphere concentric with the carrying sphere and moving about two different poles with a speed of its own. As even this was not sufficient to explain the phenomena, Eudoxus placed the poles of the second sphere on a third, which again was concentric with and larger than the first and moved about separate poles of its own, and with a speed peculiar to itself. For the planets yet a fourth sphere was required similarly related to the three others; for the sun and moon he found that, by a suitable choice of the positions of the poles and of speeds of rotation, he could make three spheres suffice. In the accounts of Aristotle the spheres are described in the reverse order, the sphere carrying the planet being the last. The spheres which move each planet Eudoxus made quite separate from those which move the others. One sphere sufficed of course to produce the daily rotation of the heavens. ‘Thus, with three spheres for the sun, three for the moon, four for each of the planets and one for the daily rotation, there were twenty-seven spheres in all. It does not appear that Eudoxus specu- lated upon the causes of these rotational motions or the way in which they were transmitted from one sphere to another; nor did he inquire about the material of which they were made, their sizes and mutual distances. In the matter of distances the only indication of his views is contained in Archimedes’ remark that he supposed the diameter of the sun to be nine times that of the moon, from which we may no doubt infer that he made their distances from the earth to be in the same ratio g: 1. It would appear that he did not give his spheres any sub- stance or mechanical connexion; the whole system was a purely geometrical hypothesis, or a set of theoretical constructions calculated to represent the apparent paths of the planets and enable them to be computed.’
Eudoxus of (Cnidus ¢. 408-355 B.c.), one of the greatest mathe- maticians of antiquity, was the discoverer of the theory of proportion expounded in the fifth book of Euclid’s Hvements and of ‘the mensura- tion of areas and volumes by the method of exhaustion, and the first
proposer of the Julian cycle. He was a pupil of Archytas and of Plato, who is said to have suggested to him for solution the problem of planetary motion (Simpl. 488. 21). He explained his system in a book On Velocities, which like all his other works is lost. Aristotle had his knowledge of the system from Polemarchus, an acquaintance of Eudoxus.
18. thy pev mpdtyy Thy Tv dmdavdv dotpwv eivar, i.e. the first (outermost) sphere of the sun (and similarly the first sphere of the moon) was meant to explain its diurnal motion from east (through south) to west. Aristotle means not that the first sphere of the sun or of the moon was the sphere of the fixed stars, but that it had the same motion.
19. Thy Sé Seutépay Kata Tov 81d péowy Tav Lwdiwv (KvKAor), i.e. the second sphere moved in the circle which bisects the signs of the zodiac longitudinally, in other words the ecliptic, the Aogds kVKos Of 107.1% 16 (which is different from the AeAofwpévos of 107320). Simplicius supposes that this second sphere produced, in the case of the moon, the revolution from west to east in a lunar month, while the third sphere produced the retrograde movement of the nodes (or points of highest latitude) in about eighteen years. But it has been pointed out that if these were the relative speeds of the two spheres ‘the moon ‘would have been found for nine years north, and then for nine years south, of the ecliptic .. . We must assume that the third sphere pro- duces the monthly revolution of the moon from west to east... round a circle inclined to the ecliptic at an angle equal to the greatest latitude of the moon, and then that this oblique circle is carried round by the second sphere in a retrograde sense along the ecliptic in a period of 223 lunations’ (Heath, 197). Simplicius’ mistake goes back to Aristotle, since ‘ Aristotle clearly implies that the second sphere cor- responds to the movement in longitude for all the seven bodies including the sun and moon, whereas in fact it only does so in the case of the five planets’ (ib.).
With regard to the sw, Simplicius says that, as in the case of the moon, the third or innermost sphere moves much more slowly than the second, but (unlike the third sphere of the moon) in the direct order of the signs (493. 15-17, 494. 6, 7, 9-11). ‘Simplicius makes the same mistake as regards the speeds of the second and third spheres as he made in the case of the moon. If it were the third sphere which moved very slowly, the sun would for ages remain in a north or a south latitude and in the course of a year would describe, not a great circle, but (almost) a small circle parallel to the ecliptic. The slow motion must therefore belong to the second sphere, the equator of which revolves in the ecliptic, while the revolution of the third sphere must take place in about a year..., the plane of its equator being inclined, at the small angle mentioned, to the plane of the ecliptic . . . The slightly inclined great circle of the third sphere which the sun appears to describe is thus carried round bodily in the
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revolution of the second sphere about the axis of the ecliptic, the nodes on the ecliptic thus moving slowly forward, in the direct order of the signs; and lastly both the second and third spheres are carried round by the revolution of the first sphere following the daily rota- tion’ (Heath, 198).
The sun’s apparent motion is, asa matter of fact, along the ecliptic, so that two circles would have been enough to explain its motion. How did Eudoxus come to suppose that it moved at a small angle to the ecliptic? Simplicius says this was inferred from the sup- posed observation that the sun, at the winter and summer solstices, does not always rise at the same point of the horizon (493. 11-17, cf. Al. 703. 27). Schiaparelli thinks that the early astronomers inferred a movement of the sun in latitude from the observed motion of the moon and the planets in latitude. This belief was opposed by Hipparchus, but lasted long; Pliny puts the inclination at one degree, Theon at half a degree. Schiaparelli (p. 17) shows that the theory was not started to explain the precession of the equinoxes, ‘ which was discovered by Hipparchus, but was unknown to Eudoxus, Pliny, and Theon’ (Heath, 200).
‘ Eudoxus supposed the annual motion of the sun to be perfectly uniform; “he must therefore have deliberately ignored the discovery, made by Meton and Euctemon sixty or seventy years before, that the sun does not take the same time to describe the four quadrants of its orbit between the equinoctial and solstitial points ’ (ib.).
23. kal tovTwy Sé Thy péev mpetyv Kal Seutépav Thy atThy etvar éxelvats, i.e. the planets shared not only the diurnal motion of the sun and the moon, but also their motion along the ecliptic. ‘The periods of this motion, ‘in the case of the superior planets, are respectively equal to the sidereal periods of revolution, and in the case of Mercury and Venus (on a geocentric system) one year. As the revolution of the second sphere was taken to be uniform, we see that Eudoxus had no idea of the zodiacal anomaly of the planets, namely that which depends on the eccentricity of their paths, and which later astronomers sought to account for by the hypothesis of eccentric circles; for Eudoxus the points on the ecliptic where successive oppositions or conjunctions took place were always at the same distances, and the arcs of retro- gradation were constant for each planet and equal at all parts of the ecliptic. Nor with him were the orbits of the planets inclined at all to the ecliptic; their motion in latitude was believed by Eudoxus to depend exclusively on their elongation from the sun and not on their longitude’ (Heath, 200-201).
26. éwd tatty, nearer than this to the centre of the universe, the earth.
27. adwacdy, sc. Tov ohaipdv Or trav dopdv. ‘Of all the planets’ would have been more accurate.
28. ris S€ tpityns dmdvtwy tods modous év TH Bid pécwy Tov Lodiov eivat, i.e. ‘the third sphere had its poles at two opposite points on the
zodiac circle, the poles being carried round in the motion of the second sphere ; the revolution of the third sphere about the poles was again uniform and took place in a period equal to the synodic period of the planet or the time which elapsed between two successive oppositions or conjunctions with the sun’ (Heath, 2or). It is not clear in which of the two possible directions this sphere rotated, but Schiaparelli shows that this does not matter for the theory.
29. THs dé TeTdpTyS Thy hopdv Kata Tov (KUKAoV Tov) Neho§wpevov mpds Tov pecov TavTyS (K’KAov). The fourth sphere moved in a circle inclined to the equator of the third. The inclination ‘was constant for each planet but different for the different planets. And the rota- tion of the fourth sphere about its axis took place in the same time as the rotation of the third about its axis but in the opposite sense. On the equator of the fourth sphere the planet was fixed, the planet thus having four motions, the daily rotation, the circuit in the zodiac, and two other rotations taking place in the synodic period’ (Heath, 201).