746 PROCEEDINGS OF THE AMERICAN ACADEMY. 



can be represented as multiples of one of them. Then if an arbitrary 

 line AB, not passing through 0, is drawn in the plane, all points on a 

 line through can be expressed as multiples of the point in which 

 this line cuts AB. Now project all points of the plane on the line AB. 

 Each point in the plane will be uniquely represented on AB if we use 

 the following convention for magnitude. Let K be any point and let 

 OK intersect AB in C. If K = AC we shall say that the projection on 

 AB of the point K is the point AC. That is, the point in which K pro- 

 jects is considered as having a magnitude X and is coincident in posi- 

 tion with C. We thus see that all points on OK will project into the 

 same point C, but the projection of each point will be looked upon as 

 having a diiferent magnitude. Now to find the sum XA + fxB we will 

 consider AA and fxB as represented vectorially and find the vector sum, 

 then project the three point AA, /xB, AA + /^B on AB. The projections 

 will have the magnitudes X, fx, \ + /x (see equation (2)) respectively. 

 If the point AA + fxB projects into (X + fx) C then for points on the 

 line AB we shall say that 



AA -h /aB = (A -f fi) C. 

 From equation (2) the point C is such that 



(Cf|AB) = -^. 



The definition of this sum is then independent of the origin chosen for 

 the vector addition. The reference element for this addition consists 

 of the line f and we have : The sum of two points AA, /xB of magnitude 

 A, fx respectively is a point (A -f /i) C of magnitude X -\- [i, dividing 



A, B in the cross ratio — ^ with respect to f. 



A 



The line f is a line of exceptional points. The sum of two distinct 

 points on this line is not defined and the sum of two coincident points 

 on f may be any given point of the plane, and therefore violates the 

 uniqueness of the sum ; besides, A times a point P on f is not necessarily 

 a point coincident in position with P. 



The magnitude of the sum was seen to be the sum of the magnitudes. 

 The difference 



A-B 



will be a point P of magnitude zero such that 



(PflAB) = l. 



and consequently P must be on f This point P is an exceptional point 

 then because it is of zero magnitude and yet is not algebraically equiva- 



