668 



SCIENTIFIC THOUGHT. 



31. 



Mutual in- 

 fluence of 

 metrical and 

 project! ve 

 geometry. 



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 projective.'' 

 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 projective 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. 



These apparent defects stimulated the representa- 

 tives of the two methods to investigate more min- 

 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 

 projected ranges), Steiner recog- 

 nised the fundamental principle 

 out of which the innumerable 

 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 



geometrical imagination capable of 

 looking at the same figure from 

 the most different sides in order 

 to multiply the number of pro- 

 perties of these curves indef- 

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

 26 ; see also Cremona, ' Projective 

 Geometry,' p. 119). 



