PHILLIPS AND MOORE, — ALGEBRA OF PROJECTIVE GEOMETRY. 



759 



and X is finite for all positions of a and b distinct from p but becomes 

 infinite when one of these lines coincides with p. 



23. Definition and properties of angle. We now define angle as a 

 scalar quantity determined by two lines not passing through F and 

 such that pairs of lines through a point give angles proportional to the 

 corresponding point vectors. We further assume that one of the lines 

 being fixed and the angle kept constant, the locus of the other line is 

 an analytic curve. That locus is then a straight line. 



Figure 12. 



Angle is thus the dual of distance. We find accordingly that there 

 exists a line f such that if two lines (not passing through F) intersect 

 on f, their angle is zero. If one of the lines passes through F, the angle 

 is indeterminate. Two lines not intersecting on f determine an angle 

 that is not zero. This angle is finite if neither of the lines passes through 

 F, but infinite if one of them passes through that point. If 



ab = k, 



the locus of b for fixed a is a point B on the line joining fa to F. The 

 correspondence between a and B is a correlation in which f and F are 

 corresponding elements and F is the locus of lines passing through their 

 corresponding points. There are two cases depending on whether F is 



