674 SCIENTIFIC THOUGHT. 



as its leader and centre, laboured at the introduc- 

 tion into pure geometry of those ideas which were 

 peculiar to the analytical method, and which gave to 

 that method its unity, generality, and comprehensiveness. 

 Two ideas presented themselves as requiring to be geo- 

 metrically dealt with : the infinite and the imaginary — 

 i.e., the elements of a figure which lie at infinity and those 

 which are ideal or invisible, which cannot be construed. 

 It is usually supposed that the consideration in geometry 

 of imaginary or invisible elements in connection with real 

 figures in space or on the plane has been imported from 

 algebra; but the necessity of dealing with tbem must 

 have presented itself when constructive geometry ceased 

 to consider isolated figures rigidly fixed, when it adopted 

 the method of referring figures to each other, of looking 

 at systems of lines and surfaces, and of moving figures 

 about or changing them by the processes of projection 

 and perspective. The analytical manipulations applied 

 to an equation, which according to some system or other 

 expressed a geometrical figure, found its counterpart 

 in projective geometry, where, by perspective methods, — 

 changing the centre or plane of projection, — certain 

 elements were made to move away into infinity, or when 

 a line that cut a circle moved away outside of it, seem- 

 ingly losing its connection with it. By such devices, 

 implying continuous motion in space, Poncelet introduced 

 and defined points, lines, and other space elements at 

 35. infinity, and brought in the geometrical conception of 



Ideal . . 



elements, ideal and imaginary elements. " Such definitions," he 

 says, " have the advantage of applying themselves at 

 once to all points, lines, and surfaces whatsoever ; they 



