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Non-Euclidean Geometry H.S.M AMS / MAA SPECTRUM VOL AMS / MAA SPECTRUM VOL 23 23 Throughout most of this book, non-Euclidean geometries in spaces of two or three dimensions are treated as specializations of real projective geometry in terms of a simple set of axioms concerning points, lines, planes, incidence, order and conti- Non-Euclidean Geometry nuity, with no mention of the measurement of distances or angles. This synthetic development is followed by the introduction of homogeneous coordinates, begin- ning with Von Staudt's idea of regarding points as entities that can be added or multiplied. Transformations that preserve incidence are called collineations. They lead in a natural way to isometries or 'congruent transformations.' Following a NON-EUCLIDEAN recommendation by Bertrand Russell, continuity is described in terms of order. Elliptic and hyperbolic geometries are derived from real projective geometry by specializing in elliptic or hyperbolic polarity which transforms points into lines GEOMETRY (in two dimensions), planes (in three dimensions), and vice versa. Sixth Edition An unusual feature of the book is its use of the general linear transformation of coordinates to derive the formulas of elliptic and hyperbolic trigonometry. The area of a triangle is related to the sum of its angles by means of an ingenious idea of Gauss. This treatment can be enjoyed by anyone who is familiar with algebra up to the elements of group theory. The present (sixth) edition clari es some obscurities in the fth, and includes a new section 15.9 on the author's useful concept of inversive distance. H.S.M. Coxeter This book presents a very readable account of the fundamental principles of hyperbolic and elliptic geometries. The approach is by way of projective geometry, the three chapters immediately following a brief historical sketch being devoted to an excellent survey of the foundations of real projective geometry. —L. M. Blumenthal, Mathematical Reviews H.S.M. COXETER AMSMAA / PRESS 4-Color Process 355 pages on 50lb stock • Spine 11/16" • Trim size 6 x 9 10.1090/spec/023 NON-EUCLIDEAN GEOMETRY Copyright, Canada, 1942, 1947, 1957 First Edition 1942 Second Edition 1947 Third Edition 1957 Fourth Edition 1961 Fifth Edition 1965 Sixth Edition 1998 By the Same Author REGULAR POLYTOPES Dover, New York THE REAL PROJECTIVE PLANE Springer-Verlag, New York INTRODUCTION TO GEOMETRY Wiley, New York PROJECTIVE GEOMETRY Springer-Verlag, New York TWELVE GEOMETRIC ESSAYS Southern Illinois University Press ©1998 by The Mathematical Association of America (Incorporated) Library of Congress Number 98-85640 ISBN 978-0-88385-522-5 e-ISBN 978-1-61444-516-6 Printed in the United States of America Current Printing: 10 9 8 7 6 5 4 3 NON-EUCLIDEAN GEOMETRY H. S. M. COXETER, C.C, F.R.S., F.R.S.C. PROFESSOR EMERITUS OF MATHEMATICS UNIVERSITY OF TORONTO SIXTH EDITION THE MATHEMATICAL ASSOCIATION OF AMERICA Washington, D.C. 20036 Brianchon's Theorem and the concurrence of angle-bisectors, (See pages 59 and 200) SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications JAMES W. DANIEL, Chair Spectrum Editorial Board ARTHUR T. BENJAMIN, Editor DANIEL ASIMOV KATHLEEN BERVER DIPA CHOUDHURY JEFFREY NUNEMACHER ELLEN MAYCOCK PARKER JENNIFER J. QUINN EDWARD R. SCHEINERMAN HARVEY J. SCHMIDT, JR. SANFORD SEGAL The Spectrum Series of the Mathematical Association of America was so named to reflect its purpose: to publish a broad range of books including biographies, accessible expositions of old or new mathematical ideas, reprints and revisions of excellent out-of-print books, popular works, and other monographs of high interest that will appeal to a broad range of readers, including students and teach- ers of mathematics, mathematical amateurs, and researchers. 777 Mathematical Conversation Starters, by John dePillis All the Math That's Fit to Print, by Keith Devlin Circles: A Mathematical View, by Dan Pedoe Complex Numbers and Geometry, by Liang-shin Hahn Cryptology, by Albrecht Beutelspacher Five Hundred Mathematical Challenges, Edward J. Barbeau, Murray S. Klamkin, and William O. J. Moser From Zero to Infinity, by Constance Reid The Golden Section, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. / Want to Be a Mathematician, by Paul R. Halmos Journey into Geometries, by Marta Sved JULIA: a life in mathematics, by Constance Reid The Lighter Side of Mathematics: Proceedings of the Eugene Sirens Memorial Conference on Recreational Mathematics & Its History, edited by Richard K. Guy and Robert E. Woodrow Lure of the Integers, by Joe Roberts Magic Tricks, Card Shuffling, and Dynamic Computer Memories: The Mathematics of the Perfect Shuffle, by S. Brent Morris The Math Chat Book, by Frank Morgan Mathematical Apocrypha: Stories and Anecdotes of Mathematicians and the Mathematical, by Steven G. Krantz Mathematical Carnival, by Martin Gardner Mathematical Circus, by Martin Gardner Mathematical Cranks, by Underwood Dudley Mathematical Fallacies, Flaws, and Flimflam, by Edward J. Barbeau Mathematical Magic Show, by Martin Gardner Mathematical Reminiscences, by Howard Eves Mathematics: Queen and Servant of Science, by E.T. Bell Memorabilia Mathematica, by Robert Edouard Moritz New Mathematical Diversions, by Martin Gardner Non-Euclidean Geometry, by H. S. M. Coxeter Numerical Methods That Work, by Forman Acton Numerology or What Pythagoras Wrought, by Underwood Dudley Out of the Mouths of Mathematicians, by Rosemary Schmalz Penrose Tiles to Trapdoor Ciphers ... and the Return of Dr. Matrix, by Martin Gardner Polyominoes, by George Martin Power Play, by Edward J. Barbeau The Random Walks of George Polya, by Gerald L. Alexanderson The Search for E.T. Bell, also known as John Taine, by Constance Reid Shaping Space, edited by Marjorie Senechal and George Fleck Student Research Projects in Calculus, by Marcus Cohen, Arthur Knoebel, Edward D. Gaughan, Douglas S. Kurtz, and David Pengelley Symmetry, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. The Trisectors, by Underwood Dudley Twenty Years Before the Blackboard, by Michael Stueben with Diane Sandford The Words of Mathematics, by Steven Schwartzman MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 800-331-1622 FAX 301-206-9789 www.maa.org PREFACE TO THE SIXTH EDITION I am grateful to the Mathematical Association of America for rescuing my book from oblivion by agreeing to publish this new edition. I particularly appreciate Elaine Pedreira's careful editing and advice. I welcome the opportunity to correct FiG. 7.2A on page 133 and to make other small improvements and especially to add the new §15.9. That section contains, in particular, an 'abso- lute' sequence of circles (FiG. 15.9A) which involves a surprising application of Fibonacci numbers (15.94). For further discus- sion of inversive geometry, the reader may like to look at "The inversive plane and hyperbolic space," Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 28 (1965), pp. 217-242. The subject of §15.7 has been extended by E. B. Vin- berg in his paper, "The volume of polyhedra on a sphere and in Lobachevsky space," American Mathematical Society Translations (2), 148 (1991), pp. 15-27. Vinberg draws attention to W. P. Thurston, "Three-dimensional manifolds, Kleinian groups and hy- perbolic geometry," Bulletin of the American Mathematical Society, 6 (1986), pp. 357-382. When Hamlet exclaims (in Act II, Scene II) "I could be bounded in a nutshell, and count myself a king of infinite space," he is providing a poetic anticipation of Poincaré's inversive model of the infinite hyperbolic plane, using a circular "nutshell" for the Absolute. In fact, the hyperbolic plane can be filled with infinitely many congruent equilateral triangles, seven at each vertex, to form the regular tessellation {3, 7}. One ac- quires an intuitive feeling for hyperbolic geometry by packing a disc with a multitude of curvilinear triangles, becoming smaller and smaller as they approach the peripheral circle, which rep- ix X PREFACE resents the Absolute. This book's cover design shows the 63 largest of these triangular tiles, systematically coloured so that any two of them which are coloured alike are entirely disjoint, having no common vertex. In other words, every vertex belongs to seven or fewer triangles, all differently coloured. H.S.M.C. June, 1998. PREFACE TO THE THIRD EDITION Apart from a simplification of p. 206, most of the changes are additions. Accordingly, it has seemed best to put the ex- tra material together as a new chapter (XV). This 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 Schäfli's remarkable formula for the differential of the volume of a tetrahedron. I gratefully acknowledge the help of L. J. Mordell and Frans Handest in the preparation of §15.6 (on quadratic forms) and §15.8 (on problems of construction), respectively. H.S.M.C. December, 1956. PREFACE xi PREFACE TO THE FIRST EDITION 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 Schläfli in Switzerland. The subject was unified in 1871 by Klein, who gave the names parabolic, hyperbolic, and elliptic to the respective systems of Euclid, Bolyai-Lobatschewsky, and Riemann-Shläfli. Since then, a vast literature has accumulated, and it is with some diffidence that I venture to add a fresh exposition. After a historical introductory chapter (which can be omitted without impairing the main development), I devote three chap- ters to a survey of real projective geometry.
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