place to place teaching the many who had freedom and inclination 

 for the work, to the Sophists, or certain of them, geometry owes 

 considerable. Chief among these was Hippocrates of Chios, who 

 was one of the first to make an attempt to solve that well-known 

 problem in mathematics the quadrature of the circle. Because, 

 however, these Sophists were not popular with the rising Platonic 

 school, the early historians have disregarded their work in great 

 measure, and as a result information in regard to them is very 

 meagre. 



Plato and his school placed likewise great importance upon 

 geometry, and it is said that over Plato's porch he had inscribed: 

 "Let no one who is unacquainted with geometry enter here." 

 Many of the definitions and some too of the axioms of Euclid are 

 to be ascribed to the Platonic school. Though it is quite true that 

 the main interest of Plato himself was in morals, yet his admiration 

 for geometry was very great. His main contribution to the science 

 however lay in this, that he trained his followers to think logically. 

 He was in mathematics a maker of mathematicians, rather than an 

 originator of discoveries. He emphasized the necessity of careful 

 definitions and distinct statements of postulates and axioms, a 

 method which Euclid afterwards adopted, and here the influence of 

 Socrates upon Plato, an influence which will be more fully ex- 

 hibited in the next chapter, is to be clearly discerned. 



The most brilliant of the mathematical pupils of Plato was 

 Eudoxus, born 408 B.C., who, it is claimed, was the author practi- 

 cally of the whole of Euclid's fifth book, and indeed it is now gener- 

 ally agreed that the substance of nearly all the geometry of Euclid's 

 Elements was known before the time of Euclid, and much of it had 

 already been methodically arranged. Upon Euclid himself there 

 devolved the task of further collecting the material, connecting it 

 where necessary, and, in some cases, enlarging the proofs. Though 

 very little is known of his life, it is certain that Euclid lived and 

 taught in Alexandria some thirty or forty years after the founding 

 of that remarkable city, the city which already had become the 

 meeting place of all the most important trade-routes, and which 

 was destined to become, throughout the next few centuries, in very 

 truth, the melting-pot for the thought of the then-known world. 

 It is hardly necessary to say anything about the nature of the 



31 



