Non-Euclidean Geometry : : Fifth Edition / / H.S.M. Coxeter.

• Superior document: Title is part of eBook package: De Gruyter University of Toronto Press eBook-Package Archive 1933-1999
• VerfasserIn: Coxeter, H.S.M.,
• Place / Publishing House: Toronto : : University of Toronto Press, , [2019]; ©1965
• Year of Publication: 2019
• Language: English
• Series: Heritage
• Physical Description: 1 online resource (326 p.)

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.

Table of Contents

Frontmatter -- PREFACE -- CONTENTS -- I. THE HISTORICAL DEVELOPMENT OF NON-EUCLIDEAN GEOMETRY -- II. REAL PROJECTIVE GEOMETRY: FOUNDATIONS -- III. REAL PROJECTIVE GEOMETRY: POLARITIES, CONICS AND QUADRICS -- IV. HOMOGENEOUS COORDINATES -- V. ELLIPTIC GEOMETRY IN ONE DIMENSION -- VI. ELLIPTIC GEOMETRY IN TWO DIMENSIONS -- VII. ELLIPTIC GEOMETRY IN THREE DIMENSIONS -- VIII. DESCRIPTIVE GEOMETRY -- IX. EUCLIDEAN AND HYPERBOLIC GEOMETRY -- X. HYPERBOLIC GEOMETRY IN TWO DIMENSIONS -- XI. CIRCLES AND TRIANGLES -- XII. THE USE OF A GENERAL TRIANGLE OF REFERENCE -- XIII. AREA -- XIV. EUCLIDEAN MODELS -- XV. CONCLUDING REMARKS -- BIBLIOGRAPHY -- INDEX