NON-EUCLIDEAN GEOMETRY 7 



showed and his methods were elaborated by Klein * that 

 the metrical properties of figures are projective properties in 

 relation to a certain fundamental figure, the Absolute, which 

 in ordinary plane geometry is a degenerate conic consisting 

 of the line infinity and the pair of imaginary points (circular 

 points at infinity) through which all circles in the plane pass, 

 but in non-Euclidean geometry is a proper conic, real in 

 Hyperbolic, imaginary in Elliptic geometry. In the language 

 of group-theory this is explained by saying that the group 

 of motions, Euclidean or non-Euclidean, is a sub-group of 

 the general projective group, and is characterised by leaving 

 invariant a certain conic. 2 



3. In ordinary plane geometry the metrical properties 

 of figures are referred to a special line, the line infinity, u, and 

 two special (imaginary) points on this line, the circular points 

 at infinity, , &/. 



The line infinity appears in point-coordinates as an equation 

 of the first degree, tt=0, while every finite point satisfies the 



geometry, but although he must be regarded as one of the epoch-makers, he never 

 quite arrived at a just appreciation of the science. In his mind non-Euclidean geometry 

 scarcely attained to an independent existence, but was always either the geometry upon 

 a certain class of curved surfaces, like spherical geometry, or a mode of representation 

 of certain projective relations in Euclidean geometry. 



1 Loc. cit., p. 5, foot-note 1. Klein has written a great deal relating to non-Euclidean 

 geometry, and was one of the first to apply it, especially in the conform representation, 

 to the theory of functions. His Erlanger Programm, Vergleichende Betrachtungen iiber 

 neuere geometrische Forschungen, 1872 (English translation in Bull. Amer. Math. Soc., 

 2 (1893) ), gives, in very condensed form, a number of representations of non-Euclidean 

 geometry, especially in relation to Lie's theory of groups. 



2 The following elementary account of the Cayley- Klein representation was published 

 in the Proc. Edinburgh Math. Soc., 28 (1910). A simple exposition from the point of 

 view of elementary geometry was given by Professor Charlotte A. Scott in the Butt. 

 Amer. Math. Soc. (2), 3 (1897). An analytical treatment is also given in her treatise on 

 Modern Analytical Geometry (London, Macmillan, 1894). The literature of this repre- 

 sentation is very extensive, as the Projective Metric, or, what comes to nearly the same 

 thing, the use of Weierstrass' coordinates (see p. 28, foot-note 1), whereby the equation 

 of a straight line is of the first degree, forms one of the most useful means of studying 

 non-Euclidean geometry. 



