68 SCIENCE IN GRECO-ROMAN ANTIQUITY 



Ancients, but modern researches have justified it. 

 Even the famous postulate concerning parallels has 

 been recognized for what it was in Euclid's conception, 

 i.e. a proposition which establishes the existence of a 

 point of intersection between two straight lines, if the 

 sum of the interior angles formed by these lines and 

 a line which cuts them be less than n. The four other 

 postulates are for the purpose of establishing the exist- 

 ence and unity of the elements needed for geometrical 

 constructions since these cannot be rigorously demon- 

 strated. The only purpose of the axioms is to set forth 

 as briefly and completely as possible the conditions of 

 equality and inequality of geometrical magnitudes. 

 These foundations once established, the geometrical 

 edifice can be constructed theorem by theorem without 

 any appeal to intuition. 



The books which form the Elements are divided 

 according to their contents as follows : I, straight lines, 

 triangles, parallelograms, the theorem of Pythagoras ; 

 II, geometrical algebra ; III the circle, angles ; IV, 

 inscribed and circumscribed polygons. These four 

 books were certainly borrowed from the Pythagorean 

 teaching, for they avoid the use of proportions even 

 when it would be most natural. 1 Book V, which treats 

 of proportions, is entirely inspired by the works of 

 Eudoxus. Book VI treats of the similitude of figures. 

 Books VII-IX make use of the works of Theaetetus 

 and treat of rational numbers, progressions, and con- 

 tinuous proportions. As to Book X (incommensurable 

 quantities) it appears to be entirely the work of Euclid. 

 In dealing with these questions, he uses the graphical 

 method, which consists in representing numbers by 

 lines, and has the advantage of providing demonstra- 

 tions applicable to all numbers, rational or irrational. 

 Books XI-XIII treat of geometry in space and are 

 inspired by Pythagoras and Plato ; they are less finished 

 1 26 Tannery, Ge'om. grecque, p. 98. 



