36 CONCRETE REPRESENTATIONS OF 



through 0. Let A' be the foot of the perpendicular from O 

 upon a, and let OA' cut the view-circle of I in Q, Q r . Then, 

 in Fig. 5, A'Q 2 =AQ*-p' 2 =AP 2 -p*+d*=c*+d*. Hence the 

 points Q, Q' depend only upon the position of the plane and 

 are independent of the line 1. Q, Q' therefore form a pair of 

 view-points for all lines in the plane. Again, for all planes 

 through I the view-points lie on the view-circle of Z, and the 



FIG. 5 



metric upon any line is the same, independently of the plane 

 in which it may be conceived to lie. 



The measure of an angle is then denned to be its apparent 

 magnitude viewed from the view-point of its plane. 



This representation is only suitable for Elliptic geometry. 

 In Hyperbolic geometry c 2 is negative, and the radius of a 

 view-circle is real only if p z > c 2 ; the view-points of a plane 

 are real only if d? > c 2 . Hence for all lines and planes which 

 do not cut the real sphere with centre O and radius ^/ c the 

 geometry is elliptic, and these lines and planes correspond 

 to ideal or ultra-spatial elements. For the lines and planes 

 which cut the sphere and which correspond to actual elements 

 the view-points are imaginary. 



