The Imaginary in Geometry 



27 



In fig. 26 the Q's are elements of the circular rays through 0. 

 Moreover, the circular rays through the center of each are tangent 

 to that one, the points of contact being / and J. The point of 

 contact of a tangent from within the black circle will always be 

 a red vector. Thus the tangent from P (fig. 26) has the point 

 of contact P 2 and is the line determined by P and P 2 . The blue 



Fig. 26. 



line from P to P.," is tangent to the hyperbola through P. 2 ". The 

 points P and P 2 ' are a pair of points inverse with respect to the 

 circle, and the distance from one to the other is double the red 

 vector which bisects that distance. Thus, in Cauchy's represen- 

 tation, arrows join every point of the plane to the inverse point 

 with respect to the circle. 



Closely related to .r 2 -(- y 2 = 1 is the purely imaginary circle 

 .v- + /-= — 1. 



27 



