An Introduction to the Geometry of N Dimensions

The present introduction deals with the metrical and to a slighter extent with the projective aspect. A third aspect, which has attracted much attention recently, from its application to relativity, is the differential aspect. This is altogether excluded from the present book. In this book a complete systematic treatise has not been attempted but have rather selected certain representative topics which not only illustrate the extensions of theorems of hree-dimensional geometry, but reveal results which are unexpected and where analogy would be a faithless guide. The first four chapters explain the fundamental ideas of incidence, parallelism, perpendicularity, and angles between linear spaces. Chapters V and VI are analytical, the former projective, the latter largely metrical. In the former are given some of the simplest ideas relating to algebraic varieties, and a more detailed account of quadrics, especially with reference to their linear spaces. The remaining chapters deal with polytopes, and contain, especially in Chapter IX, some of the elementary ideas in analysis situs. Chapter VIII treats hyperspatial figures, and the final chapter establishes the regular polytopes.