668 SCIENTIFIC THOUGHT. 



The labours of Poncelet and Steiner introduced into 

 geometry a twofold aspect, and accordingly, about the 

 middle of the century, we read a good deal of the 

 two kinds of geometry which for some time seemed to 

 develop independently of each other. The difference 

 has been defined by the terms " analytic or synthetic," 

 " calculative or constructive," " metrical or protective." 

 The one operated with formulae, the other with figures ; 

 the one studied the properties of quantity (size, magni- 

 tude), distances, and angles, the other those of position. 



The protective method seemed to alter the magnitude 

 of lines and angles and retain only some of those of 

 position and mutual relation, such as contact and inter- 

 section. The calculating or algebraical method seemed 

 to isolate figures and hide their properties of mutual 

 interdependence and relation, 

 si. These apparent defects stimulated the representa- 



Mutual in- 

 fluence of tives of the two methods to investigate more mm- 



metrical and 



utely their hidden causes and to perfect both. The 

 algebraical formula had to be made more pliable, to 

 express more naturally and easily geometrical relations ; 

 the geometrical method had to show itself capable of 

 dealing with quantitative problems and of interpreting 

 geometrically those modern notions of the infinite and 

 the complex which the analytic aspect had put promi- 



correlated theorem referring to geometrical imagination capable of 



projected ranges), Steiner recog- . looking at the same figure from 



iiised the fundamental principle j the most different sides in order 



out of which the innumerable to multiply the number of pro- 



properties of these remarkable 

 curves follow, as it were, automat- 

 ically with playful ease. Nothing 

 is wanted but the combination of 

 the simplest theorems and a vivid 



perties of these curves indef- 

 initely" (Hankel, loc. cit.. p. 

 26 ; see also Cremona, ' Projective 

 Geometry,' p. 119). 



