A THEORY OF LINEAR DISTANCE AND ANGLE. 



By H. B. Phillips and C. L. E. Moore. 

 Presented May 8, 1912, by H. W. Tyler. Received May 6, 1912. 



Introduction. 



1. In a recent article ^ we developed for the plane a theory of 

 distance and angle such that points equally distant from a fixed 

 point lie on a line and lines making a given angle with a fixed line pass 

 through a point. On account of this property we have called this 

 distance linear. In the present paper we extend this theory to higher 

 dimensions. Because of the increased complexity, the synthetic 

 method of the previous discussion cannot be used here and since we 

 know none better we have adopted that of Grassmann. In the first 

 part of the paper we have shown how the extensive quantities of 

 Grassmann can be regarded as matrices and the progressive and re- 

 gressive multiplication interpreted as simple operations performed 

 upon these matrices. In this way we develop as much of the Grass- 

 mann analysis as is needed for our purpose. We then determine for 

 any two spaces R, R' of the same dimension, a distance or angle 

 R R' having the property that if this invariant is constant and either 

 of the spaces fixed, the other satisfies a linear relation and such that 

 for three spaces R, R', R" of a pencil 



RR'+ R' R" -{- R"R = 0. 



Any distance between points that has these properties is expressible 

 in terms of a hyperplane and a linear line complex. The plane is the 

 locus of infinitely distant points and the complex the locus of minimal 

 lines. If the complex does not degenerate, the hyperplane and line 

 complex in n dimensions determine a point and 7i — 2 other complexes 

 forming altogether n elements which we use for a reference system. 

 This system of elements forms a group under outer multiplication 



1 An Algebra of Plane Projective Geometry, Proceedings of the American 

 Academy of Arts and Sciences, Vol. 47, p. 737. 



