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HISTORY OF MATHEMATICS V O L U M E 10 Sources of Hyperbolic John Stillwell AMERICAN MATHEMATICAL SOCIETY LONDON MATHEMATICAL SOCIETY Selected Titles in This Series Volume 10 John Stillwell Sources of hyperbolic geometry 1996 9 Bruce C. Berndt and Robert A. Rankin Ramanujan: Letters and commentary 1995 8 Karen Hunger Parshall and David E. Rowe The emergence of the American mathematical research community, 1876-1900: J. J. Sylvester, Felix Klein, and E. H. Moore 1994 7 Henk J. M . Bos Lectures in the history of mathematics 1993 6 Smilka Zdravkovska and Peter L. Duren, Editors Golden years of Moscow mathematics 1993 5 George W . Mackey The scope and history of commutative and noncommutative harmonic analysis 1992 4 Charles W. McArthur Operations analysis in the U.S. Army Eighth Air Force in World War II 1990 3 Peter L. Duren, editor, et al. A century of mathematics in America, part III 1989 2 Peter L. Duren, editor, et al. A century of mathematics in America, part II 1989 1 Peter L. Duren, editor, et al. A century of mathematics in America, part I 1988 Sources of Hyperbolic Geometry 10.1090/hmath/010 HISTORY OF MATHEMATICS V O LU M E 10 Sources of Hyperbolic Geometry John Stillwell AMERICAN MATHEMATICAL SOCIETY LONDON MATHEMATICAL SOCIETY Editorial Board American Mathematical Society London Mathematical Society George E. Andrews David Fowler Bruce Chandler Jeremy J. Gray, Chariman Paul R. Halmos, Chairman S. J. Patterson George B. Seligman 1991 Mathematics Subject Classification. Primary 51-03; Secondary 01A55, 53Axx, 51A05, 30F35. Library of Congress Cataloging-in-Publication Data Stillwell, John. Sources of hyperbolic geometry / John C. Stillwell. p. cm. — (History of mathematics; v. 10) Includes bibliographical references (p. - ) and index. ISBN 0-8218-0529-0 (hardcover : alk. paper) 1. Geometry, Hyperbolic— History— Sources. I. Title. II. Series. QA685.S83 1996 516.9— dc20 96-3894 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. © Copyright 1996 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. The London Mathematical Society is incorporated under royal Charter and is registered with the Charity Commissioners. 10 98765432 02 01 00 99 98 97 Preface Hyperbolic geometry is the Cinderella story of mathematics. Rejected and hidden while her two sisters (spherical and euclidean geometry) hogged the limelight, hyperbolic geometry was eventually rescued and emerged to out­ shine them both. The first part of this saga - how Bolyai and Lobachevsky laboured in vain to win recognition for their subject - is well known, and English translations of the key documents are available in Bonola’s classic Non-Euclidean Geometry. However, the turning point of the story has not been documented in English until now. Beltrami came to the rescue of hyperbolic geometry in 1868 by inter­ preting it on a surface of constant negative curvature. By giving a concrete meaning to the hyperbolic plane, he put Bolyai’s and Lobachevsky’s work on a sound logical foundation for the first time, and showed that it was a part of classical differential geometry. This was quickly followed by interpretations in projective geometry by Klein in 1871, and in the complex numbers by Poincare in 1882. Hyperbolic geometry had arrived, and with Poincare it joined the main­ stream of mathematics. He used it immediately in differential equations, complex analysis, and number theory, and its place has been secure in these disciplines ever since. He also began to use it in low-dimensional topology, an idea kept alive by a handful of topologists until the spectacular blossom­ ing of this field under Thurston in the late 1970s. Now, hyperbolic geometry is the generic geometry in dimensions 2 and 3. Alongside these developments, there has been increased interest in the work of Beltrami, Klein, and Poincare that made it all possible. I have had a steady stream of requests for the translations of Beltrami I produced in 1982, so I was delighted to be approached by Jim Stasheff with a proposal for a volume in the AMS-LMS history of mathematics series. I am also grateful to Bill Reynolds for his interest, and for help with the hyperboloid model, and to Abe Shenitzer for correcting a number of embarrassing errors in the Beltrami and Klein translations. Clayton, Victoria, Australia John Stillwell V ll Contents Introduction to Beltrami’s Essay on the interpretation of noneuclidean geometry 1 Translation of Beltrami’s Essay on the interpretation of noneuclidean geometry 7 Introduction to Beltrami’s Fundamental theory of spaces of constant curvature 35 Translation of Beltrami’s Fundamental theory of spaces of constant curvature 41 Introduction to Klein’s On the so-called noneuclidean geometry 63 Translation of Klein’s On the so-called noneuclidean geometry 69 Introduction to Poincare’s Theory of fuchsian groups, Memoir on kleinian groups, On the applications of noneuclidean geometry to the theory of quadratic forms 113 Translation of Poincare’s Theory of fuchsian groups 123 Translation of Poincare’s Memoir on kleinian groups 131 Translation of Poincare’s On the applications of noneuclidean geometry to the theory of quadratic forms 139 Index 147 IX Index additivity of measure, 74, 78 links Riemann and Poincare, angle 36 and cross-ratio, 70 Saggio, 1, 35, 72 between geodesics, 13, 14 binary quadratic forms, 118 measure, 63 Bolyai, 2, 38 of parallelism, 17, 98 Bolyai-Lobachevsky geometry, 3, 69, angle sum 115 and area, 71, 72 boundary at infinity, 117 in elliptic geometry, 96 in hyperbolic geometry, 98 catenoid, 2 universal cover, 2 in parabolic geometry, 102 Cayley, 4, 35, 69 of geodesic triangle, 19 “absolute” , 88 of spherical triangle, 96 euclidean geometry, 63 of triangle, 18, 70 measure, 72 area spherical geometry, 63 and angle sum, 71 Christoffel, 73 of geodesic polygon, 20 circle, 90 of right triangle, 19 geodesic, 12, 20, 21 Poincare definition, 129, 137 limit, 11 axiom of infinite radius, 98 line, 58 reflection, 117 parallel, 64, 111 through three points, 21 plane, 58 with centre at infinity, 24, 25 with ideal centre, 22 Battaglini, 23, 26 Codazzi, 18, 57 angle of parallelism formula, complex transformation, 131 17 congruence, 48 Beltrami, 1, 113 congruent figures, 58, 127 and Cayley formulae, 73 Poincare definition, 127, 136 compared with Klein, 64 constant curvature Fundamental Theory, 73 and Cayley measure, 73 half-plane model, 116 and the three geometries, 64 half-space model, 117 metric of Riemann, 36 letter to Genocchi, 44 space, 35, 53, 55 147 148 Index surface, 8, 55 on horosphere, 38 Cremona, 4 plane, 8 cross-ratio, 65, 124 Euler, 118 and homogeneous coordinates, 109 Fermat, 118 and right angles, 83 two square theorem, 118 harmonic, 83 Fiedler, 109 preserved by motion, 93 fixed circle, 136 von Staudt definition, 109 fixed points, 124 curvature, 8, 29 flat space, 54 as property of measure, 104 foundations of geometry, 64, 69 Gaussian, 8, 29, 53, 63, 104 and measure, 65 higher-dimensional, 72 and real numbers, 65 of measures, 63 Hilbert’s investigation, 65 spherical, 8, 29 Fuchs, 114 fuchsian functions, 113, 114 Dedekind, 41 fuchsian groups, 114, 115, 121, 123 differential equations, 114, 123 and tessellations, 118 direction, 101 fundamental region, 142 distance function and cross-ratio, 70, 79 doubly-periodic, 114 Cayley formula, 80 elliptic, 114 contrasted with angle, 74 fuchsian, 113, 114 in parabolic geometry, 102 linear fractional, 115 Klein formula, 80 modular, 125 Poincare formula, 128 periodic, 114 fundamental elliptic cone, 94 geometry, 72, 82, 95 conic, 88, 90 involution, 72 elements, 77 point, 72 rays, 82 rotation, 64 region, 142 transformation, 125, 132 surface, 69, 72 Erlanger Programm, 65 Euclid, 111 Gauss, 2, 7, 58, 119 parallel axiom, 70 and angle sum, 71 euclidean circumference formula, 16, 59, space, 27 105 euclidean geometry letter to Schumacher, 16, 105 a transitional case, 72 theorem on curvature, 8, 33 in hyperbolic geometry, 38 theory of quadratic forms, 139 in projective geometry, 63, 69 geodesic, 1, 3, 8 n-dimensional, 38 circle, 12, 20, 21, 59 Index 149 semiperimeter, 12 helicoid, 2, 24 circle with centre at infinity, Helmholtz, 69, 71 24, 25 finite space, 72 coordinate curves, 11 Hermite, 139, 141 determined by two points, 55 conjugate notation, 132 differential equation of, 29 Hilbert, 65 given by linear equation, 10, Hoiiel 41, 43, 73 translation of Beltrami, 39 in half-space model, 38 homogeneous coordinates, 63, 80 in hemisphere model, 36 derived from cross-ratio, 109 in Klein disc model, 36 homographic correspondence, 3, 33 normal bisectors, 21
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