Non-Euclidean Geometry Fifth Edition
Non-Euclidean Geometry Fifth Edition by H.S.M. Coxeter, published by University of Toronto Press on December 15, 1965, is a comprehensive exploration of geometrical systems that diverge from traditional Euclidean principles. This edition, consisting of 326 pages, delves into the historical development of non-Euclidean geometry, tracing its origins from Gauss’s initial concepts through the contributions of Bolyai, Lobatschewsky, Riemann, and Cayley. The book presents a unified view of these systems, categorized into parabolic, hyperbolic, and elliptic geometries.
Readers will find that this edition includes a new chapter that enhances the discussion with advanced topics such as mid-lines between given lines, foundational derivations in spherical and hyperbolic trigonometry, and an analysis of Gaussian curvature in various geometrical planes. The book serves as a valuable resource for those interested in mathematics and geometry, providing insights into the complexities and applications of non-Euclidean systems.
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The name non-Euclidean was used by Gauss to describe a system of geometry which differs from Euclid’s in its properties of parallelism. Such a system was developed independently by Bolyai in Hungary and Lobatschewsky in Russia, about 120 years ago. Another system, differing more radically from Euclid’s, was suggested later by Riemann in Germany and Cayley in England. The subject was unified in 1871 by Klein, who gave the names of parabolic, hyperbolic, and elliptic to the respective systems of Euclid-Bolyai-Lobatschewsky, and Riemann-Cayley. Since then, a vast literature has accumulated.
The Fifth edition adds a new chapter, which includes a description of the two families of ‘mid-lines’ between two given lines, an elementary derivation of the basic formulae of spherical trigonometry and hyperbolic trigonometry, a computation of the Gaussian curvature of the elliptic and hyperbolic planes, and a proof of Schlafli’s remarkable formula for the differential of the volume of a tetrahedron.
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