34 CONCRETE REPRESENTATIONS OF 



speak of different spaces of the same type but with different 

 space-constants. It is exactly analogous to the obvious 

 absurdity of speaking of spaces in which the total angle at a 

 point is of various magnitudes. The angle may be repre- 

 sented by different numbers, 4, 360, 6-283 . . ., and so on, 

 according to the arbitrary unit which is adopted ; so long as 

 we are dealing with one space this angle has a constant 

 magnitude, but there is no possibility of comparing magni- 

 tudes when the objects are in different spaces. There are 

 the three types of space according as K z is positive, zero, or 

 negative. For elliptic geometry, for example, different positive 

 values of K 2 mean simply a different choice of the arbitrary 

 unit of length. (Of. Russell and Whitehead, Encycl. Brit. 

 (llth ed.), article 'Geometry,' section vi., ' Non-Euclidean 

 Geometry,' p. 725d.) 



REPRESENTATION or PROJECTIVE METRIC BY APPARENT 

 MAGNITUDE AS SEEN FROM A VIEW-POINT 



27. It has been indicated in 6 that spherical and elliptic 

 geometries of two dimensions are capable of representation 

 as the geometries of a bundle of straight lines or a bundle of 

 rays through a fixed point. The former is the geometry of 

 ' visual space,' the latter is the geometry of the infinitely 

 distant elements. For a plane or a line through the fixed point 

 gives a line or a point at infinity, and the angle between two 

 lines is represented by the distance between the infinitely 

 distant points. The absolute for the geometry at infinity is 

 the imaginary circle at infinity. 



' Visual geometry ' is a two-dimensional geometry exactly 

 analogous to this except for the existence of antipodal points ; 

 i.e. as every astronomer knows, Visual geometry is the same 

 as Spherical geometry. It is the geometry which would be 

 constructed by a being endowed only with monocular vision 

 and without powers of locomotion. 



