NON-EUCLIDEAN GEOMETRY 13 



actual point make a real angle if A; is a pure imaginary, since 

 the tangents from are conjugate imaginaries. 



This then completes the representation of Hyperbolic 

 Geometry. Actual points are represented by the points 

 within a real proper conic. The conic itself consists of all the 

 points at infinity, while points outside it are ideal. 



8. If now we consider points outside the absolute as actual 

 points, there are two cases according as K is taken to be real 

 or imaginary. In the first case the distance between two 

 points will be imaginary if the line joining them does not cut 

 the absolute. Such a line must therefore be considered ideal, 

 and we get in any pencil of lines with an actual point as vertex 

 a class of ideal lines and a class of actual lines, and these are 

 separated by the two tangents to the absolute. As these 

 tangents are real, k must now be taken to be real, and we get 

 a system of angular measurement of an entirely different 

 nature from that with which we are familiar. The period of 

 the angle is now 2iirk which is imaginary, and complete 

 rotation about a point becomes impossible. If the line q is 

 a tangent to the absolute log (pq, xy) is infinite. The angle 

 between two lines thus tends to infinity as one line is rotated. 

 Further, if the line PQ touches the absolute log (PQ, XY)=0, 

 i.e. (PQ)=0, or the distance between any two points on an 

 absolute line is zero. This curious result can be found to 

 hold even in ordinary geometry if we consider imaginary 

 points. If the line PQ passes through one of the circular 

 points, so that y l y^=i(x l ~ x 2 ), then 



9. We have now to examine if the logarithmic expression 

 for the distance between two points holds in ordinary geometry. 

 In this case the two absolute points X , Y on any line PQ coin- 

 cide, and (PQ, XY)=l. The distance between any two points 

 would thus be zero if K is finite. As the distance between 

 any two points must, however, in general be finite, it follows 

 that we must make K infinite. 



