6 CONCRETE REPRESENTATIONS OF 



which points are represented by points. To a point may 

 correspond a single point or a system of points. In the latter 

 case the system of points must be regarded as a single entity, 

 and a curve which corresponds to a curve passing through a 

 point P must pass through all the points which correspond 

 to P. The representation is in fact effected by a point- 

 transformation. The straight lines of the geometry will be 

 represented by a system of curves depending upon two para- 

 meters ; and in general any curve of the system must be 

 uniquely determined when it has to pass through two distinct 

 points. In addition to such considerations, which belong to 

 analysis situs, it will be necessary also to establish the relations 

 between the metrical properties of the geometry and those 

 of its representation ; we must determine the function of the 

 positions of two points which corresponds to their distance, 

 and the function of the positions (or parameters) of two curves 

 representing straight lines which corresponds to the angle 

 between them. The distance and angle functions are not 

 independent, for a circle may be defined either as the locus of 

 a point which is equidistant from a fixed point, or as the 

 envelope of a line which makes a constant angle with a fixed 

 line, or again as the orthogonal trajectory of a pencil of lines. 



THE CAYLEY-KLEIN PROTECTIVE METRIC 



2. The simplest representation which suggests itself is 

 obtained by representing straight lines by straight lines. 

 The projective properties of non-Euclidean geometry are 

 identical with those of Euclidean geometry if we take into 

 account imaginary and infinitely distant elements. Pro- 

 jective geometry is independent of the parallel-postulate. 

 It is only in regard to metrical properties that there is any dis- 

 tinction between non-Euclidean geometry and its representa- 

 tion by the straight lines of ordinary geometry. Now Cayley J 



1 A. Cayley, 'A Sixth Memoir upon Quantics,' Phil, Trans., 149 (1859), Math. 

 Papers, vol. ii. Cayley wrote a number of papers dealing specially with non-Euclidean 



