76 A SHORT HISTORY OF SCIENCE 



elating geometry with abstract logic and philosophy, undoubtedly 

 had compensating advantages in promoting elegance and scien- 

 tific rigor, - - crystallizing out a more refined product. Archytas 

 is said also to have invented the screw and the pulley and to have 

 been the first to give a systematic treatment of mechanics, employ- 

 ing geometrical theorems for this purpose. 



MEN^ECHMUS : CONIC SECTIONS. Even more interesting in 

 its foreshadowing of future mathematical developments are the 

 solutions of the duplication problem by Mensechmus. The 

 problem which we should express in modern algebraic notation 

 by the continued proportion a : x : : x : y : :y:b, Menaechmus, with- 

 out any such notation or any system of coordinate geometry, shows 

 to be equivalent to that of determining the intersection either of a 

 parabola and a hyperbola, corresponding to the two proportions 



a : x : : x : y and a : x : :y :b, 



or to the intersection of two parabolas, in case the second pro- 

 portion is replaced by x : y : :y:b. The construction of either 

 parabola or the hyperbola naturally required some mechanical 

 device. 



The Greeks of this period distinguished three types of 

 the cone formed by the rotation of the right triangle about one 

 of its sides, according as the angle formed by that side with the 

 hypotenuse was less than, equal to, or greater than half a right 

 angle. A plane perpendicular to an element would cut a cone of 

 the first kind in an ellipse, the second in a parabola, the third in 

 a hyperbola. These curves were named accordingly sections of 

 the acute-angled, the right-angled, the obtuse-angled cone. 



The discovery of the conic sections . . . first threw open the 

 higher species of form to the contemplation of geometers. But for 

 this discovery, which was probably regarded ... as the unprofitable 

 amusement of a speculative brain, the whole course of practical phi- 

 losophy of the present day, of the science of astronomy, of the theory 

 of projectiles, of the art of navigation, might have run in a different 

 channel ; and the greatest discovery that has ever been made in the 

 history of the world, the law of universal gravitation, with its in- 



