756 PROCEEDINGS OF THE AMERICAN ACADEMY. 



corresponding line d are given, we construct b the correspondent of A 

 as follows. Draw AF and CF cutting f in P and Q and let CA cut f in 

 E,. To R corresponds the line FR. Then since three points A, C, R 

 on a line must have corresponding to them three lines through a point, 

 b, d, and FR pass through a point. Let d cut FR in K. Then PK is 

 the line b required. The construction is shown in Figure 9. 



To construct on any line through A a distance AB equal to a distance 

 CD, we construct the line b and where it cuts the line through A is the 

 point B required. If on a fixed line CD we determine a scale making 

 distances along that line proportional to the corresponding vectors 

 D — C, our construction enables us to transfer the scale to any other 

 line except f and so to assign to every pair of points, not on f, a dis- 

 tance. We must still show that this construction is consistent (if two 

 distances so constructed are equal to a third, they are equal to each 

 other) and that the resulting distance has the properties assumed in 

 the definition. 



We first show that if C and D (Figure 9) are held fixed, A and b are 

 correspondents in a correlation. The figure gives for each point A an 

 unique line b. If A moves along a line, since F and C are fixed, P and 

 R describe, on f, ranges projective with A. Also K describes on d a 

 range projective with R. Therefore P and K describe on f and d pro- 

 jective ranges. Also when A is on CF, P and R, and consequently K, 

 are at Q. Since the ranges on f and d have a self- corresponding point 

 Q, the line PK passes through a fixed point. Thus as A moves along 

 a line, b turns about a point. Furthermore, b describes a pencil projec- 

 tive with the range described by A. The correspondence between A 

 and b is therefore a correlation. 



In particular if A is on f, A and R coincide. To R therefore cor- 

 responds the line RF. Also if A is at C, b coincides with d. The 

 fact that Ab, Cd, and R, FR are pairs of corresponding elements in a 

 correlation shows that the three lines intersect in K and thus deter- 

 mines the relation between A, B and C, D. If AiBi and A2B2 are both 

 equal to CD, we have just shown that Ai, bx and A2, b2 are correspond- 

 ing pairs in a correlation which gives for a point R on f the line RF. 

 Therefore Ai, Bi and A2, B2 are related by the same kind of diagram 

 as A, B and C, D. Consequently, if two distances are by this con- 

 struction equal to a third, they are equal to each other. 



We now show that distances as constructed along any line AB are 

 proportional to the corresponding vectors B — A. We have already 

 remarked that vectors along a line preserve their ratios when projected 

 upon another line from a point on f. If we hold A and C fixed 

 (Figure 9) we may consider AB as resulting from CD by first projecting 



