Proceedings of the American Academy of Arts and Sciences. VOL. XLVIII. No. 3.-JULY, 1912. A THEORY OF LINEAR DISTANCE AND ANGLE. BY H. B. PHILLIPS AND C. L. E. Moore. 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 regressive multiplication interpreted as simple operations performed upon these matrices. In this way we develop as much of the Grassmann analysis as is needed for our purpose. We then determine for any two spaces R, R' of the same dimension, a distance or angle RR' 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 n-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. in the sense that any product of these elements is equal to a numerical multiple of a third one. In terms of this fundamental system we define the angle between any two spaces. Each of the complexes of the fundamental system is an infinite locus for spaces complimentary to it. The entire system is invariant under a group of collineations of the same order as the Euclidean group of motions. Degenerate cases are obtained by taking sections having a special relation to the fundamental system. MATRICES IN THREE DIMENSIONS. 2. Progressive Matrices. We represent a point A in three dimensions by a set of four homogeneous coordinates a,. These coordinates determine a matrix which may be used to represent the point. Two matrices of this kind will be called equal when their corresponding elements are equal. The matrix is zero if all its elements are zero. If a;=kb; we shall write A = k B. In this case the matrices A and B represent the same point but with different magnitudes. A linear function of A and B is defined by the matrix In a similar manner we define any linear function of points or matrices A, B, C, etc. If the result does not vanish it represents a point in the space determined by A, B, C, etc. If it vanishes and the coefficients are not all zero those points lie in a lower space than a like number of points usually determine. The coordinates of the line joining A and B are proportional to the two-rowed determinants in the matrix We shall call the elements of this matrix the two-rowed determinants ai ak bi bk [This is not in conformity with the usual definition which makes element equivalent to coordinate a, or b; but is the only definition which has a value in the present discussion.] The matrix is zero when all its elements are zero. In that case the points A and B have proportional coordinates and hence coincide. If the matrix is not zero it represents the line A B in the sense that from the matrix can be obtained the coordinates of the line. Conversely if the line is given a matrix can be formed by taking any two points on the line. Different matrices representing the same line are multiples of any For if A, B and P, Q are pairs of distinct points on the line one. Thus a two-rowed matrix in addition to representing a line, has a definite size. The matrix [A B] is in reality a set of six determinants The sum of two matrices [A B] and [C D] is then a complex matrix, each element of which is the sum of corresponding elements in [A B] and [CD]. In general this sum cannot be represented as a single two-rowed matrix, just as the sum of corresponding Plücker coordinates of two lines are not in general coordinates of a line. For analytical purposes we express this sum by simply writing the two matrices with an addition sign between them. If, however, the lines A B and CD intersect in a point P, we can find points Q and R on those lines such that Then [A B] = [P Q], [A B] + [CD] = [P Q] + [P R] || Pi = Pk qi+rs get re || = [P(Q+ R)]. We can consider [A B] as a product of A and B. For [A(B+C)] = [A B] + [A C] as we have just seen in the case of [P(Q + R)]. The process of multiplication consists in placing the second matrix under the matrix A |