PBINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 



281 



angle at D and the distances OC and OD. Nevertheless the fact 

 that the circle OA"C, Fig. 11, is not in absolute contact with 



A C B D 



OBDO', that the angle at E, and still more that at D, is infini- 

 tesimally in error, and that 00 . OD is not absolutely equal to 

 a 2 (or l 2 ), sufficiently establishes the purely formal nature of the 

 representation through which + oo and - oo are identified as 

 spatially the same point. In essence it is equivalent to treating 

 lines as parallel which meet at a sufficiently distant point. 



16. The projective theory of distance. — In his "Sixth Memoir on 

 Quantics," 1 Cayley developed what is generally known as a "theory 

 of distance," but what would be more readily understood if defined 

 as "a projective theory of distance," that is a theory which applies 

 not only to actual points in space, but also to their representation 

 in projections. 2 This theory was extended, simplified, and its 

 application to non-euclidean geometry pointed out by Klein. 3 The 



1 Phil. Trans., Vol. cxlix., pp 61 - 90, 1859. 



2 Space may also be conceived to vary in what may be called its 

 intensity: the theory of distance would apply with certain simple assump- 

 tions as to the law of variation of this intensity. 



3 Ueber die sogenannte nicht-euklidische Geometric Math. Amal., 

 Bd. iv., 1871. 



