750 



PROCEEDINGS OF THE AMERICAN ACADEMY, 



If 



B - A = A (D - C) 



(3) 



we might say that AB is X times as long as CD. In what follows we 

 shall assume this relation of lengths only when the points A, B, C, D 

 lie on a line. In that case A, B can be expressed as linear functions of 

 C, D, those points being assumed to be distinct and not on f Since 

 the sum of coefficients in any linear point identity must be zero, the 

 expressions will be 



Then 



Hence 



Now from (2) we have 



Figure 6. 



A = Ai C + (1 - Ai) D, 



B = A2 C + (1 — Aa) D. 



B - A = (Ai - A2) (D - C). 

 A = Aj — Aj. 



^^= (API CD) 



where P is the intersection of CD with f. Hence from the properties 

 of double ratios 



Ai = (AC|DP), 

 A2 = (BC|DP). 



