740 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



AA 2 is a point different in position from A, the addition familiar to 

 all is the well known vector addition. If AA is a point coincident 

 in position with A, but differing from it in magnitude, the sum of two 

 points is commonly defined as the centroid of the two points. We 

 shall here define these two additions in terms of projective notions. 



Vector Addition. 



6, To define the sum of two points A, B when A times a point differs 

 in position from it, we assume a line f as reference line and a point 

 as origin. Then the sum A + B is the point C determined as follows: 



Figure 1. 



Let D be the harmonic of f with respect to A, B. Then take a point 

 C so that D is the harmonic of f with respect to C, 0. The point C 

 thus obtained is the point required, and we express the relation be- 

 tween the points by the equation 



A + B = C. 



(1) 



The geometric construction for C is shown in Figure 1, and is seen to 

 be the ordinary vector construction where the line f has replaced the 

 line at infinity. 



From the above definition the point 



A + A = 2A 



* In this paper capitals will be used for points, small letters for lines. 

 Points or lines occurring as algebraic quantities will be represented by claren- 

 don type, their magnitudes by italics. Geometric points or lines may, how- 

 ever, be represented by italics when no ambiguity results. Greek letters will 

 be used for numbers. 



