60 SCIENCE IN GRECO-ROMAN ANTIQUITY 



revision of the treatise of geometry, written by Hippo- 

 crates. This revision was made first by Leon, and 

 then by Theudius of Magnesia, both pupils of the 

 Academy (Proclus, Comm. Eucl. I, pp. 66, 20 ; 67, 

 12). The trend given by Plato to astronomy was no 

 less important. His harmonious vision of the world 

 impelled him to the opinion that the irregular move- 

 ments of the planets were unreal ; preserving the 

 Pythagorean axiom of circular movement, he assigned 

 to astronomy the task of finding a combination of 

 circular movements which would account for the 

 apparent irregularity of the motion of the planets 

 (gco£ eiv Tex cpaiv /usva). 



Eudoxus of Cnidus, a contemporary of Plato, was 

 a great geometrician as well as an astronomer. Born 

 in 408 B.C., he studied under Archytas at Tarentum, 

 then he settled with his disciples at Cyzicus, which he 

 left for a time to live in Athens. He discovered 

 almost the whole of the contents of Book V of Euclid, 

 on proportions, and obtained these results by extend- 

 ing the notion of proportionality so as to include all 

 rational and irrational magnitudes. He postulates that 



— = — if ma%nb at the same time as mc^nd (m 

 a 



and n being numbers chosen arbitrarily and a, b, c, d, 

 any magnitudes) . On these foundations he established 

 the basis of the method of exhaustion, so brilliantly 

 developed by Archimedes, and which has for its com- 

 plement the reduction to absurdity. To conform to 

 the outline of astronomy sketched by Plato, he con- 

 ceived a system of homocentric spheres, the essential 

 features of which were conserved by Aristotle. It was 

 Eudoxus also who compiled the catalogue of stars, 

 used in the third century B.C. by Aratus in his poetic 

 description of the starry sky ; and it was he who 

 estimated the circumference of the earth to be 400,000 

 stadia, a value which was accepted by Aristotle. His 



