PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 



761 



direction along the line into the other is shown in Figure 14. Rotation 

 of AB around A causes B to describe the line BP. We have defined 

 AB to be the segment not crossing f. Hence, as B passes P, the seg- 

 ment AB changes into the segment AB' not crossing f (i. e. connecting 

 A to B' by way of infinity). Thus all segments into which AB can be 

 rotated lie on the same side of AF and consequently one end of the line 

 AB cannot be rotated into the other. 



Figure 14. 



A point describing (from A to B) any one of the segments AB (Figure 

 14) determines the same direction of rotation about F. Distances are 

 then positive or negative according as their description gives rotation 

 in one direction or the other around F. 



Similarly the angle ABP is the angle which does not include F. If 

 A is held fixed and B moves along BP, the angle is not changed. As 

 B passes P the angle changes into AB'P. If around B we rotate the 

 side AB into the side BP, the direction of translation (of the point of 

 intersection of AB with f ) along f is always the same. Hence an angle 

 is positive or negative according as this translation along f is in one 

 direction or the other. 



In Euclidean geometry we have an unique direction of angle about 

 any point since the equation 



angle = const., 



one side being fixed, factors into two linear conditions and so gives a 

 separation of classes, but no definite direction along a line since 



distance = const., 



