DEVELOPMENT OF MATHEMATICAL THOUGHT. 667 



30. 



It must, however, in all fairness be stated that about 

 the period from 1822 to 1830 this 'great simplification 

 and unification of geometric science was as it were in the 

 air that it had presented itself to various great thinkers 

 independently, being suggested from different points of 

 view. The beginnings can no doubt be traced in the 

 beautiful theorems of older French mathematicians, such 

 as Pascal and De la Hire, and more generally in the 

 suggestive methods of Monge and Poncelet ; its first 

 formal enunciation is in the memoirs of Gergonne : but 

 the comprehensive use of it the rewriting of geometry 

 from this point of view was the idea of Jacob Steiner, steiner 

 who, in his great but unfinished work on the " Systematic 

 Development of the Dependence of Geometric .Forms " 

 (1830), set himself the great task "of uncovering the 

 organism by which the most different forms in the world 

 of space are connected with each other." " There are," 

 he says, " a small number of very simple fundamental 

 relations in which the scheme reveals itself, by which 

 the whole body of theorems can be logically and easily 

 developed." " Through it we come, as it were, into pos- 

 session of the elements which Nature employs with the 

 greatest economy and in the simplest manner in order to 

 invest figures with an infinite array of properties." l 



rebut, de leur substituer des traites 

 d'une forme tout-a-fait differente, 

 des trait^s vraiment philosophiquea 

 qui nous montrent enfin cette ^ten- 

 due, receptacle universel de tout ce 

 qui existe, sous sa veritable physi- 

 onomie, que la mauvaise mdthode 

 d'enseignement adopted jusqu'a ce 

 jour ne nous avait pas perinis de 

 remarquer ; il s'agit, en un mot, 

 d'opeVer dans la science une revolu- 



tion aussi impe'rieusement n^ces- 

 saire qu'elle a ete jusqu'ici peu 

 preVue." 



1 See the Preface to the 'Sys- 

 tematische Entwickelung, &c. ,' 

 in Jacob Steiner's ' Gesammelte 

 Werke' (ed. Weierstrass), vol. i. 

 p. 229. " In the beautiful theorem 

 that a conic section can be gener- 

 ated by the intersection of two 

 projective pencils (and the dually 



