PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 



741 



is the point on the line OA such that A is the harmonic of f with re- 

 spect to 2A, 0. The point 



2A + A = 3A 



from the definition is constructed as follows: Let C be the harmonic 

 of f with respect to 2A, A and then construct the point 3A so that C 

 is the harmonic of f with respect to 3A, 0. It is at once seen from 



3A 



the construction of 2 A that C is the point — - . From the theory of 



cross ratios it is also seen that the relation between 3A, A is expressed 



by the cross ratio 



(0,f|3A, A) = 3. 



In like manner the points AA for integral values of A are constructed. 

 If X is the reciprocal of an integer /*, the point AA can be constructed 

 thus: Take a point B not on the line OA and construct the points 



B, 2B, 3B, fiB. 



Figure 2. 



Connect /iB to A and let the line cut f in P. Draw a line from the point 

 B to P. Where PB cuts A is the point required, for the same harmo- 

 nic relations hold among the A's as among the B's. The construction 

 of the points AA for rational values of A is now evident. For irra- 

 tional values of A the construction is obtained by limiting processes. 

 From the theory of cross ratios it is seen that the relation between A 

 and AA is expressed by the cross ratio 



