XXXI V 



admirable memoir* ' Ueber die sogenannte Nicht-Euklidische Geo- 

 metrie,' which contains a considerable simplification in statement of 

 Cayley's original point of view, and contributes one of the most im- 

 portant results of the whole theory. The work of the two mathe- 

 maticians now being an organic whole, there is no advantage at 

 least here in attempting to subdivide the subject for the purpose of 

 specifying the exact share of each in its construction. 



The scope of the Cayley-Klein ideas may briefly be gathered from 

 the following sketch. Let A, and A 2 be two points, often called a 

 point-pair ; they are to be either both real or, if not both real, then 

 conjugate imaginaries so far as their co-ordinates are concerned. 

 Let P, Q, ft be three other points on the line A t A 2 ; and let the 

 symbol (PQ) denote 



according as A! and A 2 are a real point-pair, or an imaginary point-pair. 

 Then it is manifest that 



(PQ) + (QR) = (PR), 



so that the functions (PQ), (QR), (PR) satisfy the fundamental 

 property of the distances between P and Q, Q and R, and P and R. 

 Consequently (PQ) may be taken as a generalised conception of the 

 distance between the points P and Q. 



Now let a conic be described in a plane, either imaginary, say, of the 

 form x 2 -\-y* + z z = or real, say, of the form x* + y 2 z* = 0. Choosing 

 the latter case, let attention be confined to points lying within the 

 conic, so that every straight line through a point cuts the conic in a 

 real point-pair. Take two points, P and Q ; and let the line joining 

 them cut the conic in two points, AI and A 2 . Then (PQ), as defined 

 above (the constant 7 being the same for all such lines), is the gene- 

 ralised distance between P and Q. This conic, which has been 

 arbitrarily assumed, and upon which the generalised conception of 

 distance depends, is termed by Cayley the Absolute. 



Cayley, however, avoided the unsatisfactory procedure of using 

 one conception of distance to define a more general conception. As he 

 himself explains more fully, f he regarded the co-ordinates of points 

 as some quantities which define the relative properties of points, con- 

 sidered without any reference to the idea of distance but conceived as 

 ordered elements of a manifold. Thus if a,, /3 l5 7! and 2 , /3 2 , 7a be the 

 co-ordinates of the point-pair A! and A 2 , the co-ordinates of the points 



* ' Math. Ann.,' vol. 4 (1871), pp. 573625. 



t See the note which he added, ' C. M. P.,' vol. 2, p. 604, to the Sixth Memoir ; 

 it contains some interesting historical and critical remarks. 



