Geometry of Four Dimensions

Abstract

THIS work, which is primarily a formal treatise on the differential geometry of curves, surfaces, and threefold regions in ordinary homaloidal space of four dimensions, follows fairly closely on the lines of the author's very well known treatise on differential geometry in a three-dimensional space. Prof. Forsyth naturally prefers, however, in a treatise of this kind, not to assume, on the part of the reader, any previous knowledge of the elementary metrical properties of lines, planes, and planar threefolds in four dimensions, and he devotes the first half of his first volume, therefore, to a very systematic investigation of such properties, partly as a foundation to the subsequent theory, and partly for their own sake. This section of his work includes, therefore, a consideration of such concepts as the four direction cosines of a line and the six orientation co-ordinates of a plane, leading up to a very complete set of formulae for the inclination and other properties of lines, planes, and threefolds in four dimensions; and though some readers will probably claim that part of this work could have been simplified by making use of the projective aspect of metrical four-space, and in particular by introducing the concepts of solid at infinity and absolute quadric, yet no one will deny that the author's treatment is very clear and adequate as it stands. He concludes this more elementary work by a chapter on what he calls globular representation of directions, analogous to spherical representation in ordinary space, and uses this in the analysis of general finite or infinitesimal displacements of the orthogonal frame of reference.

Geometry of Four Dimensions.

Prof. A. R. Forsyth. Vol. 1. Pp. xxix + 468. Vol.2. Pp. xi + 520. (Cambridge: At the University Press, 1930.) 75s. net.