PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 763 



ure 15) determine the same cross ratios with triads of lines through A. 

 In fact, let 



It was shown in § 1 that 



ABi = AB2 = 1. 

 A (ABO + H- (AB2) 



Figure 15. 



intersects B1B2 in a point H such that 



(HBi I B2P) = 



M 



26. To add two segments we have moved them to the point of in- 

 tersection of their lines. If the two lines intersect on f, this is not pos- 

 sible. We shall now derive a construction for the sum of two segments 

 not at the same point. We have seen that equal vectors along a line 

 (also equal segments on a line) project from a point on f into equal 

 vectors on any other line. Since the sum of segments at a point is 

 determined by harmonic constructions involving only those segments 

 and f, it follows that addition relations connecting segments through a 

 point are projective from a point on f Any sum of segments may be 

 found by a succession of processes consisting of moving two of the seg- 

 ments to the point of intersection of their lines and then adding them 

 vectorially. Hence any linear relation connecting segments holds for 

 the projections of those segments upon any line from a point on f 



