says that Eudoxus "first increased the number of general theorems, added to the three proportions three more, and raised to a considerable quantity the learning, begun by Plato, on the subject of the section, to which he applied the analytical method." By this 'section' is meant, no doubt, the "golden section" (sectio aurea), which cuts a line in extreme and mean ratio. The first five propositions in Euclid XIII. relate to lines cut by this section, and are generally attributed to Eudoxus. Eudoxus added much to the knowledge of solid geometry. He proved, says Archimedes, that a pyramid is exactly one-third of a prism, and a cone one-third of a cylinder, having equal base and altitude. The proof that spheres are to each other as the cubes of their radii is probably due to him. He made frequent and skilful use of the method of exhaustion, of which he was in all probability the inventor. A scholiast on Euclid, thought to be Proclus, says further that Eudoxus practically invented the whole of Euclid's fifth book. Eudoxus also found two mean proportionals between two given lines, but the method of solution is not known.
Plato has been called a maker of mathematicians. Besides the pupils already named, the Eudemian Summary mentions the following: Theætetus of Athens, a man of great natural gifts, to whom, no doubt, Euclid was greatly indebted in the composition of the 10th book,[8] treating of incommensurables; Leodamas; Neocleides and his pupil Leon, who added much to the work of their predecessors, for Leon wrote an Elements carefully designed, both in number and utility of its proofs; Theudius of Magnesia, who composed a very good book of Elements and generalised propositions, which had been confined to particular cases; Hermotimus of Colophon, who discovered many propositions of the Elements and composed some on loci; and, finally, the names of Amyclas of Heraclea, Cyzicenus of Athens, and Philippus of Mende.
