78 A SHORT HISTORY OF SCIENCE 



rems was expressed in the auxiliary theorem: "When two 

 volumes are unequal, it is possible to add their difference to 

 itself so many times that the result shall exceed any assigned 

 finite volume." This exceedingly useful and important princi- 

 ple, avoiding the difficulties of infinitesimals, was expressed in 

 several approximately equivalent forms, and was already im- 

 plied in the work of Antiphon and Bryson. A solution of the 

 duplication problem which gained Eudoxus the appellation 

 "godlike" has been entirely lost. 



There appear to have been no astronomical instruments at this 

 time except the simple gnomon and sun-dial, but the more 

 obvious irregularities of the planetary motions were beginning to 

 attract attention, and under Eudoxus led to the development of 

 a new and important theory. Nearest to the central earth is 

 the moon, carried on the equator of a sphere revolving from west 

 to east in 27 days. The poles of this sphere are themselves car- 

 ried on a second sphere, which turns in about 18| years about 

 the axis of the zodiac. The angle between the axes of these two 

 spheres corresponds with the moon's variation in latitude. A 

 third outer sphere gives the daily east to west motion. Simi- 

 larly there are three spheres for the sun. For each of the five 

 planets a fourth sphere is necessary to account for the stations 

 and retrogressions of its apparent orbital motion thus making 

 with the single sphere of the stars 27 spheres, all having their 

 common centre at the centre of the earth. 



How far these spheres were regarded as having concrete exis- 

 tence, how far they merely expressed in convenient geometrical 

 form the observed relations and motions, we cannot determine 

 from extant evidence. The amount of observational data available 

 was entirely inadequate to serve as a basis for any quantitatively 

 correct theory. The third sphere of the sun was based on an 

 erroneous hypothesis as to its motion. For Mercury, Jupiter and 

 Saturn the theory was reasonably adequate, for Venus less so, and 

 for Mars quite defective. 



Calippus, a follower of Eudoxus, endeavored with some degree 

 of success to remedy these defects by adding a fifth sphere for 



