KEY FEATURES:
 Some representative topics which illustrate the extensions of three dimensional geometry.
 No treatise on Ndimensions.
 Projective aspect is discussed with ideas relating to algebraic varieties and account of quadrics with reference to linear spaces.
 Metrical aspects give, in addition to Cartesian formulae, some accounts and applications of the Pliicher–Grassmann coordinates of a linear space and applications to linegeometry.
 Polytopes are discussed in detail leading to regular polytopes.
 References are of original works.

ABOUT THE BOOK: The present book 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 rather a selected certain representative topics have been discussed which not only illustrate the extension of theorems of threedimensional geometry, but also 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 5 and 6 are analytical, the former projective, the latter largely metrical. In the former are given some of the simplest ideas relating to lgebraic 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 9, some of the elementary ideas in analysis situs. Chapter 8 treats hyperspatial figures and the final chapter establishes the regular polytopes.

ABOUT THE AUTHOR(S): D M Y Sommerville (1879–1934) was a Scottish mathematician and astronomer. He compiled a bibliography on NonEuclidean geometry and also wrote a leading textbook in that field. He was a cofounder and the first secretary of the New Zealand Astronomical Society. In 1910, Duncan wrote “Classification of geometries with projective metrics”. Sommerville was elected a Fellow of the Royal Society of Edinburgh in 1911.
In 1915, Sommerville went to New Zealand to take up the Chair of Pure and Applied Mathematics at the Victoria College of Wellington. Duncan became interested in honeycombs and wrote “Division of space by congruent triangles and tetrahedra” in 1923. The following year he extended results to ndimensional space.
He also discovered the DehnSommerville equations of the number of faces of convex polytopes. Sommerville used geometry to describe the voting theory of a preferential ballot. He addressed Nanson’s method where n candidates are ordered by voters into a sequence of preferences. Sommerville showed that the outcomes lay in n ! simplexes that covered the surface of an n– 2 dimensional spherical space. In 1926, he became a fellow of the Royal Astronomical Society.



CONTENTS:
Fundamental Ideas ParallelsPerpendicularityDistances and Angles between Flat SpacesAnalytical Geometry: ProjectiveAnalytical Geometry: MetricalPolytopes Mensuration ContentEuler’s TheoremThe Regular Polytopes
