THE HELLENIC PERIOD 61 



disciple Menaechmus was equally remarkable. The 

 tutor of Alexander the Great, he replied to a question 

 of his royal pupil by saying that there are no royal 

 roads in geometry. 1 Following the suggestions of 

 Archytas, he resolved the problem of the duplication 

 of the cube by finding the point of intersection of either 

 the two parabolas x 2 = ay, y 2 = 2ax, or the parabola 

 x 2 = ay and the hyperbola xy = 2a 2 . These equations 

 result directly from the mean proportionals enunciated 

 by Archytas and Hippocrates. 



a x y 



x y 2a 



Menaechmus may have shown besides that these curves 

 can be obtained by the intersection of a plane and a 

 cone of revolution, and thus opened up the way for the 

 theory of conic sections. 



7. ARISTOTLE AND THE PERIPATETIC 

 SCHOOL. THE NATURAL SCIENCES 



Aristotle (384-322 B.C.) directed the study of 

 science into new paths. The son of a physician, he 

 was as much interested in natural science and inductive 

 methods as in metaphysics and exact science. He was 

 at first a disciple of Plato, but he left the Academy after 

 the death of his master. The writings he has left are 

 valuable and varied. The greater part have come down 

 to us in the form of notes written for an oral exposition, 

 and they constitute a veritable encyclopaedia of the 

 knowledge of the period. But Aristotle not only col- 

 lected, systematized, and discussed the opinions of his 

 predecessors and contemporaries, he created entirely 

 new systems such as logic, morphology, and biological 

 classifications. It must be noted, however, that 

 although he had sufficient mastery of elementary 

 mathematics to use them as illustrations of his logic, 



1 This saying is also attributed to Euclid, 



