PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 



749 



we see also that AC cuts BD on f. Thus the relation of the four 

 points is shown by the diagram (Figure 5). 



Figure 5. 



Conversely, if the points A, B, C, D have the positions shown in the 

 figure, f has the same harmonic with respect to A, D and B, C, or 



Consequently, 



A + D = B + C. 

 B - A = D - C. 



If the four points lie on a line cutting f in P, we project, irom some 

 other point on f, the points A, B to M, N on a second line through P. 

 Then from the intersection of MC and f we project N to D' on AB. 

 By construction 



B-A = N-M = D'-C 

 Hence, if 



B - A = D - C, 



the points D and D' must coincide. The relation of the four points is 

 shown in Figure 6. 



The quantity B — A has properties very similar to those of the 

 vector AB in the ordinary vector analysis. In fact, the expressions 

 B — A add exactly like vectors if the line f is at infinity. On account 

 of this similarity we shall use the term vector to indicate the quantity 

 B-A. 



