28 A Short History of Astronomy [Cn. u. 



to scientific Greek astronomy. As in the schemes of 

 several of his predecessors, the fixed stars lie on a sphere 

 which revolves daily about an axis through the earth ; the 

 motion of each of the other bodies is produced by a com- 

 bination of other spheres, the centre of each sphere lying 

 on the surface of the preceding one. For the sun and 

 moon three spheres were in each case necessary : one to 

 produce the daily motion, shared by all the celestial 

 bodies ; one to produce the annual or monthly motion in 

 the opposite direction along the ecliptic ; and a third, with 

 its axis inclined to the axis of the preceding, to produce 

 the smaller motion to and from the ecliptic. Eudoxus 

 evidently was well aware that the moon's path is not 

 coincident with the ecliptic, and even that its path is not 

 always the same, but changes continuously, so that the third 

 sphere was in this case necessary ; on the other hand, he 

 could not possibly have been acquainted with the minute 

 deviations of the sun from the ecliptic with which modern 

 astronomy deals. Either therefore he used erroneous 

 observations, or, as is more probable, the sun's third sphere 

 was introduced to explain a purely imaginary motion con- 

 jectured to exist by "analogy" with the known motion of 

 the moon. For each of the five planets four spheres were 

 necessary, the additional one serving to produce the variations 

 in the speed of the motion and the reversal of the direction of 

 motion along the ecliptic (chapter i., 14, and below, 51). 

 Thus the celestial motions were to some extent explained 

 by means of a system of 27 spheres, i for the stars, 6 for 

 the sun and moon, 20 for the planets. There is no clear 

 evidence that Eudoxus made any serious attempt to arrange 

 either the size or the time of revolution of the spheres so as 

 to produce any precise agreement with the observed motions 

 of the celestial bodies, though he knew with considerable 

 accuracy the time required by each planet to return to the 

 same position with respect to the sun ; in other words, his 

 scheme represented the celestial motions qualitatively but 

 not quantitatively. On the other hand, there is no reason 

 to suppose that Eudoxus regarded his spheres (with the 

 possible exception of the sphere of the fixed stars) as 

 material ; his known devotion to mathematics renders it 

 probable that in his eyes (as in those of most of the 



