THE GOLDEN AGE OF GREECE 77 



numerable direct and indirect consequences and applications to every 

 department of human research and industry, might never to this hour 

 have been elicited. Sylvester. 



Many of Plato's followers and disciples in the Academy con- 

 tinued the development of mathematics. To Xenocrates, for 

 example, is attributed the determination of the number of all 

 possible syllables as 1,002,000,000,000, a result obtained by 

 some unknown method. This whole period is one of great pro- 

 ductivity and importance in the history of mathematics. New 

 theorems and new methods are discovered, former methods 

 are critically scrutinized, loci problems are investigated, these 

 and the study of the three classical problems leading to the 

 introduction of new curves and a general extension of geo- 

 metrical knowledge. Geometry, with emphasis, indeed, on its 

 philosophical side, predominates over the theory of numbers, 

 and even the latter is given so geometrical a form that mathe- 

 matics is unified. 



A NEW COSMOLOGY. Eudoxus of Cnidos (408 ?-355 B.C.) 

 was a student both of Archytas and, for a time, of Plato. He was 

 not only mathematician and astronomer, but also physician. In 

 mathematics he is almost a new creator of the science, developing 

 the theory of proportion, making a special study of the "golden 

 section," already mentioned in connection with the regular poly- 

 gons, and obtaining important results in solid geometry. In the 

 words of the register, "Eudoxus of Cnidos . . . first increased 

 the number of general theorems, added to the three propor- 

 tions three more, and raised to a considerable quantity the learn- 

 ing begun by Plato on the subject of the (golden) section, to which 

 he applied the analytical method." 



To him was formerly attributed the proof that the volume of 

 a pyramid is one third that of the prism having the same base 

 and altitude, as well as the corresponding theorem for cones and 

 cylinders. A recently discovered manuscript of Archimedes shows, 

 however, that for this Democritus deserves the credit. The 

 method of exhaustion, so-called, employed in proving these theo- 



