The combined effect of the rotation of the third and fourth spheres is thus described by Simplicius (496. 23—497. 6): ‘ The third sphere, which has its poles on the great circle of the second sphere passing through the middle of the signs of the zodiac, and which turns from south to north and from north to south, will carry round with it the fourth sphere which also has the planet attached to it, and will more- over be the cause of the planet’s movement in latitude. But not the third sphere only ; for, so far as it was on the third sphere (by itself), the planet would actually have arrived at the poles of the zodiac circle and would have come near to the poles of the universe ; but, as things are, the fourth sphere, which turns about the poles of the inclined circle carrying the planet and rotates in the opposite sense to the third, i.e. from east to west, but in the same period, will prevent any considerable divergence (on the part of the planet) from the zodiac circle, and will cause the planet to describe about this same zodiac circle the curve called by Eudoxus the Azppopede, so that the breadth of this curve will be the (maximum) amount of the apparent deviation of the planet in latitude, a view for which Eudoxus has been attacked’ (Heath, 201-202). Schiaparelli has shown how it was possible for Eudoxus, with the geometrical knowledge at his command, to arrive at the Azppopede (horse-fetter) or spherical lemniscate (a sort of figure of eight) as the path of a planet so far as it is determined by the third and the fourth of its spheres. But in virtue of the second gopd the lemniscate itself moves along the ecliptic. The actual motion of the planet among the fixed stars is due to the combination of these two motions. For half the synodic period the motion of the planet along the lemniscate accelerates its motion along the ecliptic, and for half of the period it retards it. When the backward motion along the lemniscate is greater than the forward motion of the lemniscate the planet retrogrades, and when the two motions are equal it is stationary. The theory is evidently meant to explain the retro-
390 Commentary
gradations and the stations of the planets; while the breadth of the lemniscate defines their motions in latitude. Except in the case of Mars, Eudoxus (according to the figures given by Simplicius 495. 26-29, 496. 6-9) assigned fairly accurately both the synodic and the zodiacal or sidereal periods, which implies the use of careful observa- tions whether Egyptian or Babylonian. -We~do not know the angles of inclination of the axis ef the fourth to that of the third sphere which he assigned for the several planets, but taking the most probable angles Schiaparelli has shown that ‘for Jupiter and Saturn, and to some extent for Mercury also, the system was capable of giving on the whole a satisfactory explanation of their motion in longitude, their stationary points, and their retrograde motions; for Venus it was unsatisfactory, and it failed altogether in the case of Mars. The limits of motion in latitude represented by the various /zppopedes were in tolerable agreement with observed facts, although the periods of the deviations and their places in the cycle were quite wrong’ (Heath, 211).
30. elvar Sé THs TpiTHs ohalpas Tos médous TOV pev GANwv idious, tos S€é THs “Appoditys kal Tod ‘Eppod Tods adtovs. ‘As regards Mercury and Venus, inasmuch as their mean positions coincide with the mean position of the sun, Eudoxus must have assumed that the centre of the hippopede always coincides with the sun. ‘This centre being on the ecliptic and at a distance of go° from each of the poles of rotation of the third sphere, the poles of the third sphere of Mercury and the poles of the third sphere of Venus coincide’ (Heath, 210).
31-35. These names for the planets are apparently late. They occur first in Pl]. Epznoms 9878 f., where they are mentioned as compara- tively new (the name Hermes occurs in Z?m.'38D). Plato ascribes the names to a Syrian origin, and they were in fact derived from Babylonia. In earlier Greek literature only “Eozepos and “Ewodopos are mentioned by name, though the names ®aivwy (Saturn), @acOwv (Jupiter), TIvpdes (Mars), ®woddpos (Venus), Sr/ABwv are probably old (Burnet, £. G. P.2 23, n. 1).
32. Callippus of Cyzicus (fl. 330 B.c.) studied with Polemarchus, a friend of Eudoxus, and is said to have stayed at Athens with Aristotle, ‘correcting and completing, with Aristotle’s help, the discoveries of Eudoxus’ (Simpl. 493. 5-8).
33-34. tod . . . ta, which is omitted by E, is doubtless a gloss like those (also beginning with rodr’ éo7v) which A has in A. 984? 11, T. r009% 26. Cf. I. 1053» 31,
34. TO peév Tod Ards kal TO TOO Kpdvou Td adtd exeivw drediSov. As a matter of fact, Eudoxus’ theory, as we have seen, works best for these planets. Callippus had evidently ‘not perceived the elliptic inequality in the motion of either planet, though it can reach the value of five or six degrees ’ (Dreyer, 104). Norcan he have perceived their deviations in latitude.
35- TOS FrLw kal TH ceAyvy S00 Weto Ett mpocOereas civar ohaipas,
TG Horvdpeva, ei pedder Tus dmoddcew. Simplicius tells us that ‘accord- ing to Eudemus, Callippus asserted that, assuming the periods between the solstices and equinoxes to differ to the extent that Euctemon and Meton held that they did, the three spheres in each case (i.e. for the sun and moon) are not sufficient to save the phenomena, in view of the irregularity which is observed in their motions’ (Heath, 218). With regard to the sun, Euctemon, about 430 B.c., ‘had made the length of the seasons (beginning with the vernal equinox) 93, 90, 90, and g2 days respectively . . . Callippus, about 330 B.c., made the correspond- ing lengths 94, 92, 89, 90 days respectively’ (ib. 215)—a much more accurate estimate. Callippus accounted for the inequality by supposing, besides the three spheres attributed by Eudoxus to the sun, a fourth with its poles on the third, and a fifth with the sun on its equator, its poles on the fourth sphere, and its axis slightly inclined to the axis of the fourth; the fifth sphere rotating at the same speed as the fourth and in the opposite direction. Thus Callippus explains the sun’s unequal motion in longitude as Eudoxus explained the synodic inequalities of the planets, by a Azppopede; and ‘this representation of the motion of the sun is almost as accurate as that obtained later by means of the eccentric circle and the epicycle’ (id. 216). Sim- plicius implies that Callippus assigned two new spheres to the moon for the same reason; i.e. he ‘was aware of the inequality in the motion of the moon in longitude’ (ib.)—a discovery which would naturally have resulted from comparing the times of lunar eclipses with the corresponding longitudes of the moon. Here again a Azppopede would explain all the facts except evection.
37- Tots S€ Norrois TOv ThayyTwv ExdoTw piav. Simplicius tells us that ‘the reason why Callippus added the one sphere which he added in the case of each of the three planets Ares, Aphrodite,and Hermes was shortly and clearly stated by Eudemus’ (497. 17-24); but he does not tell us what it was. We have already seen that Eudoxus’ system fails signally with Mars. ‘The fifth sphere was probably meant to account for the retrogradations of Mars, without assuming as Eudoxus did a synodic period other than the true one (260 instead of 780 days). Schiaparelli has been able to show how three concentric spheres instead of Eudoxus’ latter two will give the planet at certain points ‘a much greater direct and retrograde velocity with the same motion in latitude’ (Heath, 215) and thus ‘preserve the appearances’ much better.
In‘ the case of Venus and Mercury also, Callippus’ fifth sphere enabled him to approach nearer to the facts than Eudoxus had done.
38. Eudoxus and Callippus had offered a purely geometrical account of the planetary system; Aristotle aimsat a mechanical account, and can- not isolate the system of one planet from that of the next. He therefore supposes for each ‘ planet’ except the moon certain spheres which ‘ roll back’ the outer sphere of the planet just nearer to the earth than the given planet, i.e. which prevent the influence of the forward-moving or
392 Commentary
deferent spheres of one planet from affecting the next. The mode of operation of the ‘ backward-rolling ’ spheres is explained clearly by Heath. ‘Suppose A, B, C, D to be the four spheres postulated for Saturn, A being the outermost and D the innermost on which the planet is fixed. If inside the sphere D we place a first reacting sphere D’ which turns about the poles of D with~equal speed, but in the opposite sense, to D, the rotations-of D and D’ will mutually cancel each other and any point of D’ will move as though it was rigidly con- nected with the sphere C. Again, if we place inside the sphere D’ a second reagent sphere C’ rotating about the same poles with C and with equal speed, but in the opposite sense, the rotations of C and C’ cancel each other, and any point of C’ will move as if it were rigidly connected with the sphere B, Lastly, if inside C’ a third reagent sphere B’ is introduced which rotates about the same poles with B and at the same speed but in the opposite sense, the rotations of B and B’ will cancel each other and any point of B’ will move as if it were rigidly con- nected with the sphere A. But, as A is the outermost sphere for Saturn, A is the motion of the sphere of the fixed stars; hence B’ will move in the same way as the sphere of the fixed stars; and consequently Jupiter's spheres can move inside B’ as if the spheres of Saturn did not exist and as if B’ itself were the sphere of the fixed stars’ (p. 218). In this system, however, both the innermost reacting sphere of a planet and the next sphere to it, the outermost deferent sphere of the next planet, are moving with the same motion, viz. that of the fixed stars, so that the second of these two spheres is superfluous. Aristotle might thus have reduced the total number of spheres by six.
107425. The subject of movetoOar is daravra, which = ovyrebetoar maga. (at opaipa) 1073" 38. tiv dopdv answers to Ta hawopeva 1074°T,
6. at pev dxrd, i. e. four each for Saturn and Jupiter (1073 23, 34).
7. at dé mévTe Kat elkoow, i.e. five for each of the other five bodies (107317 and 35, 23 and 37).
7-8. tovtwv dé wdvas 06 Set... Pepetar. ‘Aristotle should have realized that, strictly speaking, the account which he gives in the Me/eorologica of shooting stars, comets, and the Milky Way necessitates the introduction of four reacting spheres below the moon. For, according to Aristotle, these phenomena are the effects of exhalations rising to the top of the sublunary sphere and there coming into contact with another warm and dry substance which, being the last layer of the sublunary sphere and in contact with the revolution of the outer heavenly sphere, is carried round with it; the rising exhalations are kindled by meeting and being caught in the other substance and are carried round with it. Hence there must be a sphere below the moon which has the same revolu- tion as that of the sphere of the fixed stars, in order that comets, &c., may be produced and move as they are said to do. The four inner spheres producing the moon’s own motion should therefore be neutra- lized as usual by the same number of reacting spheres’ (Heath, 219).
0 ee ee ee Ns ee Cee eit nee ees eae
A. 8. 1074 5-14 393
10-12, 6 8}... wévte. The number of the spheres in the several
theories is as follows :
Eudoxus Callippus Aristotle
Saturn 4 4 7
Jupiter 4 4 7
Mars 4 5 9
Venus 4 5 9
Mercury 4 5 9
Sun 3 5 9
Moon 3 5 5
26 33 55