PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 745 



at the intersection of the line joining P to A' + B' + C with the line 

 joining Q to A" + B" + C", which is independent of the way in which 

 A, B, C are combined. 



9. For the dual of the above addition we assume a point 0' as 

 reference point, and a line f as reference line. Then from duality we 

 see that the lines Aa all pass through the same point on f. The sum 

 a + b is formed by drawing a line c harmonic of 0' with respect to 

 a, b. Then drawing the line d so that c is the harmonic of 0' with 

 respect to f ', D, the relation between c, d is 



d = 2 c. 



The relations between Aa, fxh, a, b are the same as in the case of 

 points. That is, the line h joining the point of intersection off and 

 Aa + /*b with the point of intersection of a, b is such that 



(hO'iab)^-^ (1) 



Aa + ^b = (A + fx) h. (2) 



The reference elements 0', f need not be the same as 0, f. How- 

 ever, if the addition of points and its dual are to be used together, the 

 line joining AA, AB should be X times the line joining A, B. The line 

 joining AA to AB intersects the line AB on f, for XA and AB (for vari- 

 able A) are two projective ranges with as self-corresponding point, 

 and consequently the ranges are perspective. The line f joins corre- 

 sponding points (corresponding to the infinite value of X) and therefore 

 lines joining pairs of corresponding points must pass through the same 

 point on f Then if the line joining A A, AB is to be A times AB the 

 line f must coincide with f The dual argument will show that 0' 

 should coincide with 0. The fundamental or reference system for this 

 vector addition then consists of a line f and a point (). 



The line f is exceptional in the addition for there is no way shown of 

 finding the sum of two distinct points on this line. 



Point Addition. 



10. In this case AA 3 is coincident in position with A but differs from 

 A in magnitude. The number A indicates the magnitude of XA. In 

 the case of vector addition we saw that all points on a line through 



^ Throughout the discussion of point addition points denoted by A, B, C, . . . 

 will always be understood to be unit points. 



