738 PROCEEDINGS OF THE AMERICAN ACADEMY. 



by the fact that AA is in general distinct from A. In the point addition 

 the sum of two points A and B is the harmonic of the singular line with 

 respect to those points. For this addition A.A is in general coincident 

 in position with A. The two additions are closely related. In fact, each 

 is representable in terms of addition processes of the other kind. When 

 the singular line is taken at infinity these additions agree essentially 

 with those of Grassmann. 



2. In the case of point addition A and XA have the same position. 

 According to Mobius these quantities differ in weight. To give a geo- 

 metric interpretation to this weight we conceive a point as a sort of 

 double fan-shaped spread consisting of all the lines through the point 

 and between two limiting lines. The size of the point is then measured 

 by the angle (directed) between these limiting lines, and the weight of 

 the point is this angle. We are thus led to define a species of angle in 

 which the total angular magnitude about a point is infinite. The finite 

 angle determined by two lines is that one which does not contain the 

 singular point. 



In the same way we represent a line by a segment; of itself and the 

 magnitude of the line by the length of that segment. We thus define 

 a sort of distance in which the locus of point at a fixed distance from a 

 given point is a straight line. This distance between two points is the 

 dual of the angle between two lines. It has a definite algebraic sign 

 and along each line not passing through the singular point assigns a 

 definite positive direction. 



Distance and angle are invariant under a three-parameter group of 

 collineations (projectively equivalent to motions leaving area invariant) 

 for which the singular point and line are fixed elements. These colline- 

 ations leave invariant the correlations having the fixed point and line 

 as coincidence loci. There are two cases depending on whether the 

 fixed point is on the line or not. The first of these gives a distance 

 theory similar to that in a minimum plane. The second does not occur 

 as a special case of distance defined relative to a conic. 



3. In terms of this linear distance and angle there is a very simple 

 theory of the triangle. Most rational relations of ordinary trigonometry 

 involving distances and sines of angles are replaced by similar relations 

 involving distances and angles. In case the singular point is on the 

 singular line there is a linear relation between the sides and also between 

 the angles of a triangle making both distance and angle similar to angle 

 in ordinary geometry. If the point is not on the line, however, any 

 three parts determine the triangle. Every part is then rationally ex- 

 pressible in terms of any three, and similar triangles do not exist. 



In this system there is no right angle, and distance from point to line 



