THE HELLENIC PERIOD 59 



circle. Without labouring the point, we see how 

 fruitful the work of this geometrician was. 



Archytas of Tarentum followed it up in the dupli- 

 cation of the cube; he pointed out a very elegant 

 method of discovering the mean proportionals, a 

 method which implies a very clear understanding of 

 " geometrical loci." According to Archytas the two 

 mean proportionals sought are obtained by the inter- 

 section of the three following surfaces : 

 the cylinder x 2 + y 2 = ax 



a 2 

 the cone x 2 + y 2 +z 2 — —x 2 



the tore or anchor-ring (x 2 + y 2 + z 2 ) 2 = a 2 (x 2 + y 2 ) 

 this latter being produced by the revolution of a circle 

 around one of its tangents. 1 



As for Plato (427-347 b.c.) we know the value he 

 attached to mathematics. He borrowed from it the 

 basis of his idealism, since mathematical demonstration 

 cannot be based upon the observations of sensible 

 phenomena, for Nature displays only imperfect figures. 

 On the other hand, this demonstration could not be 

 arbitrarily created by the mind. There exists there- 

 fore beyond the realm of sensible perception a realm 

 of ideas of which our minds gradually become aware. 

 Thus scepticism and sensualism are checked. Without 

 making any real discoveries, Plato has defined the 

 conditions of mathematical research. He insists on 

 the necessity of reducing axioms and definitions to 

 the smallest number possible. He distinguishes be- 

 tween the analytical method by which one can ascer- 

 tain if the problem be solvable or not, and the synthetic 

 method by which the solutions are worked out. In 

 this way Plato rendered invaluable service as much in 

 the research of primary propositions as in the con- 

 struction of geometrical figures. His advice led to a 



1 28 Tannery, Mem. set., II, p. 19. 



