DEVELOPMENT OF MATHEMATICAL THOUGHT. 665 



Poncelet's method of central projection attacked geometri- 

 cal problems from a purely constructive point of view. 

 Nevertheless the frequently expressed oljject of the later 

 writings of Monge, as well as those of Carnot and 

 Poncelet, was to introduce into geometrical reasoning 

 the generality and continuity which analysis possessed, 

 and this was largely attained by the interpretation of 

 notions taken over from analysis. Their endeavours 

 were, however, in the sequel crowned by the discovery 

 of a purely geometrical property, the understanding of 

 which has ever since formed the basis of what may be 

 termed modern geometry. 



This remarkable property, which may be regarded 

 as revealing the very essence of extension in space or 

 of the " space -manifold," — inasmuch as it brings the 

 different elements of space into mutual relation, — is the 

 go-called principle of " duality " or of " reciprocity." The ss. 



. , I J Principle ol 



principle of duality is now usually defined to mean that duality. 

 in geometry on the plane or in space, " figures coexist in 

 pairs, two such coexisting figures having the same genesis 

 and only differing from one another in the nature of the 

 generating element." ^ The elements of plane geometry 

 are the point and the line ; the elements of solid geometry 

 i are the point and the plane. By interchanging these 

 I correlative terms, correlative propositions may be written 

 down referring to plane and to space geometry. In pro- 

 jective geometry there are two processes which are cor- 

 ; i relative or complementary to each other — the process of 

 ;i projection and the process of section. We can project 



^ Cremona, ' Elements of Projective Geometry,' transl. by Leudesdorf. 

 Oxford, 1885, p. 26. 



