662 SCIENTIFIC THOUGHT. 



of motion of points, lines, planes — corresponded accord- 

 ingly to the notion of vai-iability in analysis. The intro- 

 duction of motion, gradual and continuous, would give 

 to purely geometrical or descriptive reasoning the same 

 flexibility which analysis had acquired in the calculus of 

 fluxions and of variations. Figures would lose their 

 rigidity and isolation and limited nature and become 

 movable, related to each other, filling the whole of space 

 24. instead of a restricted and confined area or region. It 



Character 



of modem is the peculiarity of the modern as opposed to the older 



geometry. j. ^ x x 



geometry, never to let figures become motionless or 

 rigid,^ never to consider them in their isolation, but 

 always in their mutual relations ; never to have regard 

 only to a finite portion of a line or surface, but to 

 conceive of it in its infinite extension. By a reaction 

 of analysis and geometry on each other, freedom and 

 generality have been gradually acquired. 



But this moving about of figures in space in order 

 to learn their properties and mutual relations must b& 

 according to some method ; otherwise it will not lead to 

 scientific and exact knowledge. Poncelet, in considering 

 how the two successful methods in geometry — the 

 Cartesian and the Descriptive — had attained to their 

 perfection, discovers a general principle which underlies 

 their proceedings, and which is capable of great extension: 

 this is the principle of projection." 



^ See, inter ulia, what Geiser 

 says of Jacob Steiner's method in 

 his pamphlet ' Zur Erinnerung an 

 Jacob Steiner,' Schaffhausen, 1874, 

 p. 27. 



^ 'Traite des Proprietes pro- 

 jectives,' voh i. p. xviii : " En 

 reflechissant attentivement Ji ce 



qui fait le principal avantage de 

 la Geometric descriptive et de la 

 mdthode des coordonnees, h ce qui 

 fait que ces branches des Mathe- 

 matiques offrent le caractere d'une 

 veritable doctrine, dont les prin- 

 cipes, peu nombreux, sont li^s et 

 enchaiiies d'une maniere necessaire 



