THE GOLDEN AGE OF GREECE 75 



There is absolutely nothing in his various statements about 

 the construction of the universe tending to show that he had de- 

 voted much time to the details of the heavenly motions, as he never 

 goes beyond the simplest and most general facts regarding the revo- 

 lutions of the planets. Though the conception of the world as Cos- 

 mos, the divine work of art, into which the eternal ideas have breathed 

 life, and possessing the most godlike of all souls, is a leading feature 

 in his philosophy, the details of scientific research had probably no 

 great attraction for him, as he considered mathematics inferior to 

 pure philosophy in that it assumes certain data as self-evident, for 

 which reason he classes it as superior to mere opinion but less clear 

 than real science. 



Through his widely read books he helped greatly to spread the 

 Pythagorean doctrines of the spherical figure of the earth and the 

 orbital motion of the planets from west to east. 



The conjunction of philosophical and mathematical activity 

 such as we find, beside Plato, only in Pythagoras, Descartes and 

 Leibnitz, has always borne the finest fruits for mathematics. To the 

 first we owe scientific mathematics in general. Plato discovered the 

 analytical method, through which mathematics was raised above the 

 standpoint of the Elements, Descartes created analytic geometry, our 

 own celebrated countryman Leibnitz the infinitesimal calculus, - 

 and these are the four greatest steps in the development of mathe- 

 matics. - Hankel. 



ARCHYTAS. To Archytas, a late Pythagorean, with whom 

 Plato had had close relations, was due the earliest solution of the 

 duplication problem. This very interesting and somewhat elab- 

 orate solution involves a combination of three services, a cone of 

 revolution, a cylinder having the vertex of the cone in the cir- 

 cumference of its base, and a surface generated by revolving a 

 semicircle about an axis passing through one end of its diam- 

 eter. It shows remarkable mastery of elementary geometry, 

 both plane and solid, and an interesting tendency to employ 

 a wider range of methods, including motion, which might, but 

 for adverse tendencies, have had important results in connecting 

 mathematics with its possible applications to mechanics, etc. 

 The influence of Plato in avoiding such connections and asso- 



