NON-EUCLIDEAN GEOMETRY 35 



28. A somewhat analogous representation for geometry 

 of three dimensions has been devised by E. M'Clintock and 

 modified by W. W. Johnson. 1 



We have seen that the geometry on the surface of a sphere 

 gives, by central projection on any plane, a representation by 

 straight lines with the Cayley-Klein projective metric. On 

 every plane, with the exception of those through the centre 

 of the sphere, a definite metric is thus established. To 

 eliminate these exceptional planes M'Clintock proceeds in 

 this way. A fixed point is taken in space, and the metric 

 on any plane through this point is defined to be that upon a 

 tangent plane to the sphere in which corresponds to the 

 point of contact. The metric upon any other plane at a 

 distance r from is then defined to be that upon a plane 

 parallel to the tangent plane, and at a distance r from it on the 

 opposite side from the centre, the foot of the perpendicular 

 from corresponding to the foot of the perpendicular from 

 the centre of the sphere. 



This procedure is modified in an elegant manner by John- 

 son. Assume a ' central point ' and a linear magnitude c 

 corresponding to the radius of the sphere ; then the projective 

 measure of a segment is its apparent magnitude viewed from 

 a point P at a distance c from measured in a direction 

 perpendicular to the plane through the given line and 0. All 

 lines in this plane have the same view-point, or pair of view- 

 points. 



Consider any line ?, and let the plane through perpendi- 

 cular to I cut I in A. Draw a circle with centre A passing 

 through P, P'. Any point on this circle will also be a view- 

 point for the line I. Hence a line has a view-circle. 



Consider any plane a, and take a line I in it. Construct 

 the view-circle of I, whose centre is A and whose plane passes 



1 E. M'Clintock, ' On the non-Euclidean Geometry,' New York, Bull. Amer. Math. 

 Soc., 2 (1892), 21-33. W. W. Johnson, ' A Case of non-Euclidean Geometry,' ibid., 

 158-161. 



