DEVELOPMENT OF MATHEMATICAL THOUGHT. 673 



can be applied also to geometry in space, the point 

 being conceived as generating a plane by its motion, or 

 three planes defining a point by their intersection, leads 

 us to the same idea of dual correspondence or reciprocity 

 which Poncelet and Gergonne had arrived at by entirely 

 different considerations. Pliicker's was an analytical 

 mind, and with him the principle of duality at once 

 assumes an analytical form. He saw that the same 

 equation lent itself to a twofold interpretation, accord- 

 ingly, as we adopt point co-ordinates or line co-ordinates 

 i.e., according as we refer our geometrical figure to the 

 point or the line as the moving and generating space 

 element. Through this step the idea of co-ordinates 34. 



Generalised 



was generalised, and the dualistic conception of figures co-ordin- 

 in space received an analytical expression. It was 

 the junction of analytical and descriptive methods on a 

 higher level, from which an entirely novel and fertile 

 development of geometry became possible. 



Whilst the labours of Pliicker lay in the direction of 

 making analytical formulae more natural, better adapted 

 to the expression of geometrical forms and relations, and 

 of reading out of these remodelled formulae novel geometri- 

 cal properties, the French school, with Michel Chasles 1 



1 In addition to numerous valu- 

 able, memoirs, Chasles published, 

 among others, two works of para- 

 mount importance, inasmuch as 

 they for a long time dominated 

 purely geometrical research, not 

 only in France but also in Ger- 

 many and England, the 'Aper^u 

 historique sur 1'origine et le 

 deVeloppement des me"thodes en 

 ge'ome'trie' (1837), and the 'Traite" 

 de ge'ome'trie supeYieure ' (1852). 

 These works, through their bril- 



liant style, not only threw into the 

 shade for a time the labours of 

 contemporary German mathema- 

 ticians, such as Mobius, Steiner, 

 Pliicker, and Von Staudt, but also 

 obscured some of the single dis- 

 coveries of the author himself. 

 The ' Aper9u ' was early trans- 

 lated into German ; whereas in 

 this country it was the Dublin 

 school, notably Townsend and Dr 

 Salmon, who spread a knowledge 

 of Chasles's work. 



VOL. II. 2 U 



