THE HELLENIC PERIOD 



55 



Geometry, on the contrary, made rapid progress. 

 Theodorus of Cyrene enunciated the problem of 



the incommensurables V3, V5, etc., up to V17 

 (Plato, Theaetetus, 147, D). Three problems especially 

 attracted attention, for although they present them- 

 selves as the natural generalization from simple 

 geometrical constructions, yet they cannot be directly 

 solved by the means of the rule and compass. These 

 three problems, fa- 

 mous in the history 

 of mathematics, 

 are : the trisection 

 of the angle, the 

 quadrature of the 

 circle, the duplica- 

 tion of the cube. 1 

 They gave rise to 

 numerous and fruit- 

 ful investigations, 

 and gradually led 

 to the theory of 

 conic sections. The 

 primary impulse 

 was given by the 

 sophists. Hippias 

 of Elis first dis- 

 covered the curve called the quadratrix. This 

 curve (Fig. 5) is obtained by the intersection of 

 the moving radius of a circle and a straight line which 



1 The duplication of the cube is also called the Deliac prob- 

 lem. Apollo, having been consulted about the plague which 

 ravaged Athens in 430 B.C., directed that, in order to end it, 

 the volume of the altar of Delos, which was cubical, should be 

 doubled. The Athenians thought to do this by simply doub- 

 ling the sides of the altar ; but, the scourge having redoubled, 

 they recognized their error and applied to Plato. Aristotelis 

 opera, IV, p. 209, scholies de Philipon aux Analytiques pos- 

 terieures. 

 5 



Fig. 5. 



