DEVELOPMENT OF MATHEMATICAL THOUGHT. 675 



are, besides, neither indifferent nor useless, they help to 

 shorten the text and to extend the object of geometrical 

 conceptions ; lastly, they establish a point of contact, if 

 not always real, at least imaginary, between figures which 

 appear -prvma, vista to have no mutual relation, and 

 enable us to discover without trouble relations and 

 properties which are common to them." l It was the 

 principle of geometrical continuity which led Poncelet 

 to the consideration of infinite and imaginary elements. 



As we saw above, the protective methods of Poncelet 

 had introduced into geometrical reasoning a remark- 

 able distinction among the properties of figures. In 

 general it was recognised that, in the methods of 

 central and parallel projection or in drawing in per- 

 spective, certain properties or relations of the parts of 

 a figure remain unaltered, whereas others change, be- 

 come contorted or out of shape. Poncelet called the 

 former projective or descriptive, the latter metrical, 

 properties. This distinction introduced into all geom- 

 etry since his time several most important and funda- 

 mental points of view ; it divided geometrical research 

 into two branches, which we may term positional 

 and metrical geometry the geometry of position and 

 that of measurement. We know that ancient geometry 

 started from problems of mensuration : modern geometry 

 started, with Monge, from problems of representation or 

 graphical description. It has thus become a habit to 

 call ancient geometry metrical, modern geometry pro- 

 jective. This habit has led to an unnecessary separation 

 of views, but in the further course of development also 



1 ' Trait^ des Proprtet^s projectives,' vol. i. p. 28. 



