764 



PROCEEDINGS OP THE AMERICAN ACADEMY. 



This gives us a means of constructing the sum of two segments AB 



and CD. Let DA and BC cut f in P and Q. The sum AB + CD lies 



on some line LM through the intersection of AB and CD. From P 



(Figure 16) AB + CD is projected upon any line into a segment limited 



by the lines PC, PB. Likewise from Q it projects into a segment 



limited by QA, QD. Let PC intersect QA in L, and PB intersect QD 



in M. Then 



AB + CD = LM. 



-f5» 



M 



Figure 16. 



27. Just as we represent a line by a segment (sequence of points 

 between two given points) so we represent a point by a sector (sequence 

 of lines between two given lines). The sector used is the one no line 

 of which passes through F. A point of magnitude A is represented by 

 a sector of angle A. 



We conceive that the points in a discussion are thus replaced by 

 fan-shaped spreads (the lines extending on both sides of the point). 

 This spread, or sector, may be rotated about its vertex without chang- 

 ing its value by merely keeping the angle constant. Two of these 

 sectors may be added vectorially after turning them around until the 

 initial lines are the same. Let the sectors then be ab and ac and let 

 the harmonic of F with respect to b and c be d. Then 



ab -I- ac = 2ad. 



The sum of any two sectors may be found at once by a construction 

 dual to that for adding segments (Figure 17). 



