NON-EUCLIDEAN GEOMETRY 15 



and it appears as in the analogous case just considered that 

 there is now no natural unit of angle available, as the period is 

 infinite. A unit must be chosen conventionally. 



The geometries in the case in which k is infinite or real 

 present a somewhat bizarre appearance, and are generally on 

 that account excluded from discussion, the objection being 

 that complete rotation about a point is impossible, and the 

 right angle has no real existence. Yet, if we go outside the 

 bounds of plane geometry, such geometries will present 

 themselves when we consider the metrical relations subsisting 

 on certain planes, ideal or at infinity. 



Let us consider the case of hyperbolic geometry of three 

 dimensions. Here the absolute is a real, not ruled, quadric 

 surface, say an ellipsoid, and actual points are within. Actual 

 lines and planes are those which cut the absolute, and the 

 geometry upon an actual plane is hyperbolic. But an ideal 

 plane cuts the absolute in an imaginary conic, and the geometry 

 upon such a plane is elliptic. A tangent plane to the absolute 

 cuts the surface in two coincident points and a pair of imagin- 

 ary lines. The geometry on such a plane is the reciprocal of 

 Euclidean geometry, i.e. the measurement of distances is 

 elliptic while angular measurement is parabolic. In this 

 geometry the perimeter of a triangle is constant and equal to 

 IT, just as in Euclidean geometry the sum of the angles is 

 constant and equal to TT. Now if we make use of the theorem 

 that the angle between two planes is equal to the distance 

 between their poles with respect to the absolute, we see that 

 the geometry of a bundle of planes passing through a point 

 on the absolute is Euclidean. The sum of the three dihedral 

 angles of three planes whose lines of intersection are parallel 

 is therefore always equal to TT, a result which was obtained by 

 Lobachevsky and Bolyai. 1 



1 A complete classification of all the geometries arising from the Cayley-Klein 

 representation in space of n dimensions will be found in the author's paper, ' Classifica- 

 tion of Geometries with Projective Metric,' Proc. Edinburgh Math. Soc., 28 (1910). 



