Plato (427-347 B. C.) was probably not an original mathematician, but he 'caused mathematics in general and geometry in particular to make a great advance by reason of his enthusiasm for them'. He encouraged the members of his school to specialize in mathematics and astronomy; e. g. we are told that in astronomy he set it as a problem to all earnest students to find 'what are the uniform and ordered movements by the assumption of which the apparent motions of the planets may be accounted for'. In Plato's own writings are found certain definitions, e. g. that of a straight line as 'that of which the middle covers the ends', and some interesting mathematical illustrations, especially that in the second geometrical passage in the _Meno_ (86E-87C). To Plato himself are attributed (1) a formula _(n²-1)²+(2n)²=(n²+1)²_ for finding two square numbers the sum of which is a square number, (2) the invention of the method of analysis, which he is said to have explained to Leodamas of Thasos (_mathematical_ analysis was, however, certainly, in practice, employed long before). The solution, attributed to Plato, of the problem of the two mean proportionals by means of a frame resembling that which a shoemaker uses to measure a foot, can hardly be his. Eudoxus (408-355 B. C.), an original genius second to none (unless it be Archimedes) in the history of our subject, made two discoveries of supreme importance for the further development of Greek geometry. (1) As we have seen, the discovery of the incommensurable rendered inadequate the Pythagorean theory of proportion, which applied to commensurable magnitudes only. It would no doubt be possible, in most cases, to replace proofs depending on proportions by others; but this involved great inconvenience, and a slur was cast on geometry generally. The trouble was remedied once for all by Eudoxus's discovery of the great theory of proportion, applicable to commensurable and incommensurable magnitudes alike, which is expounded in Euclid's Book V. Well might Barrow say of this theory that 'there is nothing in the whole body of the elements of a more subtile invention, nothing more solidly established'. The keystone of the structure is the definition of equal ratios (Eucl. V, Def. 5); and twenty-three centuries have not abated a jot from its value, as is plain from the facts that Weierstrass repeats it word for word as his definition of equal numbers, and it corresponds almost to the point of coincidence with the modern treatment of irrationals due to Dedekind.