H ISTO RY OF M ATH EM AT ICS VO LU 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. Moore1994

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 . McArthurOperations 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

HISTORY OF MATHEMATICS

VOLUME 10

Sources of Hyperbolic Geometry

John Stillwell

AMERICAN MATHEMATICAL SOCIETY

LONDON MATHEMATICAL SOCIETY

10.1090/hmath/010

Editorial BoardAmerican Mathematical Society

George E. Andrews Bruce Chandler Paul R. Halmos, Chairman George B. Seligman

London Mathematical SocietyDavid FowlerJeremy J. Gray, Chariman S. J. Patterson

1991 Mathematics Subject Classification. Primary 51-03; Secondary 01A55, 53Axx, 51A05, 30F35.

Library of Congress Cataloging-in-Publication DataStillwell, 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 1996516.9—dc20 96-3894

CIP

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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 out­shine 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 inter­preting 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 main­stream 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 blossom­ing 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’sEssay on the interpretation of noneuclidean geometry 1

Translation of Beltrami’sEssay on the interpretation of noneuclidean geometry 7

Introduction to Beltrami’sFundamental theory of spaces of constant curvature 35

Translation of Beltrami’sFundamental theory of spaces of constant curvature 41

Introduction to Klein’sOn the so-called noneuclidean geometry 63

Translation of Klein’sOn the so-called noneuclidean geometry 69

Introduction to Poincare’sTheory of fuchsian groups, Memoir on kleinian groups,On the applications of noneuclidean geometryto the theory of quadratic forms 113

Translation of Poincare’sTheory of fuchsian groups 123

Translation of Poincare’sMemoir on kleinian groups 131

Translation of Poincare’sOn the applications of noneuclidean geometryto the theory of quadratic forms 139

Index 147

IX

Index

additivity of measure, 74, 78 angle

and cross-ratio, 70 between geodesics, 13, 14 measure, 63 of parallelism, 17, 98

angle sumand area, 71, 72 in elliptic geometry, 96 in hyperbolic geometry, 98 in parabolic geometry, 102 of geodesic triangle, 19 of spherical triangle, 96 of triangle, 18, 70

areaand angle sum, 71 of geodesic polygon, 20 of right triangle, 19 Poincare definition, 129, 137

axiomline, 58parallel, 64, 111 plane, 58

Battaglini, 23, 26angle of parallelism formula,

17Beltrami, 1, 113

and Cayley formulae, 73 compared with Klein, 64 Fundamental Theory, 73 half-plane model, 116 half-space model, 117 letter to Genocchi, 44

links Riemann and Poincare, 36

Saggio, 1, 35, 72 binary quadratic forms, 118 Bolyai, 2, 38Bolyai-Lobachevsky geometry, 3, 69,

115boundary at infinity, 117

catenoid, 2universal cover, 2

Cayley, 4, 35, 69 “absolute” , 88 euclidean geometry, 63 measure, 72 spherical geometry, 63

Christoffel, 73 circle, 90

geodesic, 12, 20, 21 limit, 11of infinite radius, 98 reflection, 117 through three points, 21 with centre at infinity, 24, 25 with ideal centre, 22

Codazzi, 18, 57 complex transformation, 131 congruence, 48 congruent figures, 58, 127

Poincare definition, 127, 136 constant curvature

and Cayley measure, 73 and the three geometries, 64 metric of Riemann, 36 space, 35, 53, 55

147

148 Index

surface, 8, 55 Cremona, 4 cross-ratio, 65, 124

and homogeneous coordinates, 109

and right angles, 83 harmonic, 83 preserved by motion, 93 von Staudt definition, 109

curvature, 8, 29as property of measure, 104 Gaussian, 8, 29, 53, 63, 104 higher-dimensional, 72 of measures, 63 spherical, 8, 29

Dedekind, 41differential equations, 114, 123direction, 101distance

and cross-ratio, 70, 79 Cayley formula, 80 contrasted with angle, 74 in parabolic geometry, 102 Klein formula, 80 Poincare formula, 128

ellipticgeometry, 72, 82, 95 involution, 72 point, 72 rotation, 64transformation, 125, 132

Erlanger Programm, 65 Euclid, 111

parallel axiom, 70 euclidean

space, 27euclidean geometry

a transitional case, 72 in hyperbolic geometry, 38 in projective geometry, 63, 69 n-dimensional, 38

on horosphere, 38 plane, 8

Euler, 118

Fermat, 118two square theorem, 118

Fiedler, 109 fixed circle, 136 fixed points, 124 flat space, 54foundations of geometry, 64, 69

and measure, 65 and real numbers, 65 Hilbert’s investigation, 65

Fuchs, 114fuchsian functions, 113, 114 fuchsian groups, 114, 115, 121, 123

and tessellations, 118 fundamental region, 142

functiondoubly-periodic, 114 elliptic, 114 fuchsian, 113, 114 linear fractional, 115 modular, 125 periodic, 114

fundamental cone, 94 conic, 88, 90 elements, 77 rays, 82 region, 142 surface, 69, 72

Gauss, 2, 7, 58, 119 and angle sum, 71 circumference formula, 16, 59,

105letter to Schumacher, 16, 105 theorem on curvature, 8, 33 theory of quadratic forms, 139

geodesic, 1, 3, 8circle, 12, 20, 21, 59

Index 149

semiperimeter, 12 circle with centre at infinity,

24, 25coordinate curves, 11 determined by two points, 55 differential equation of, 29 given by linear equation, 10,

41, 43, 73in half-space model, 38 in hemisphere model, 36 in Klein disc model, 36 normal bisectors, 21 on sphere, 9 parallels, 15, 54, 57 polygon, 20 sphere, 59 triangle, 15, 19, 55

formulae, 18 geometry

Bolyai-Lobachevsky, 115 elliptic, 63, 82, 95 euclidean, 63 foundations, 64 hyperbolic, 63, 96 parabolic, 63, 99 projective, 63 pseudo-, 142 pseudospherical, 58 relative consistency, 64 spherical, 58, 62, 63

Godel’s theorem, 65 group

discontinuous action, 119 fuchsian, 114, 115, 121, 123 kleinian, 116, 118, 131 linear, 137 modular, 119 of isometries, 117 of linear fractional transforma­

tions, 117of substitutions, 114 of transformations, 65, 92 symmetry, 118

helicoid, 2, 24 Helmholtz, 69, 71

finite space, 72 Hermite, 139, 141

conjugate notation, 132 Hilbert, 65 Hoiiel

translation of Beltrami, 39 homogeneous coordinates, 63, 80

derived from cross-ratio, 109 homographic correspondence, 3, 33 homographic transformation, 140 homography, 50 horocycle, 2, 26, 59

formula of Lobachevsky, 26 superposition of, 2, 4, 26

horocyclic disc, 2 horocyclic sector, 4 horosphere

euclidean geometry on, 38 n-dimensional, 38

hyperboliccoordinates, 141 geometry, 4, 72, 96

n-dimensional, 38 involution, 72 plane, 4

in hyperbolic space, 117 point, 21, 72 rotation, 64 space, 38

and Klein disc, 38 in euclidean space, 38 motion, 117 n-dimensional, 38 tessellation of, 118

transformation, 126, 132 hyperboloid model, 120

idealcentre of geodesic circle, 22 centre of rotation, 98 point, 22

150 Index

imaginary circle at infinity, 70 imaginary circular points, 76, 99,

106imaginary points, 72 inversion

in a circle, 117, 133 in a sphere, 117, 133 in unit circle, 117 preserves hyperbolic distance,

117isometry, 116

as product of inversions, 117 of hyperbolic space, 117

isomorphic groups, 124

Klein, 4, 63, 113, 125, 137 disc model, 3, 35

and hyperbolic space, 38 Erlanger Programm, 65

kleinian groups, 116, 118, 131

Lagrange, 114, 118reduction of forms, 119

Lame, 28 lattice, 118 Legendre, 70 length

and linear fractional functions, 115

measure, 63Poincare definition, 116, 128,

136limit

circle, 11, 14 curve, 59of hyperbolic space, 44 surface, 59

lineaxiom, 58 element

euclidean, 27of pseudospherical surface, 21

hyperbolic, 116

pencil, 82 postulate, 9

for geodesics, 9 linear equations

and lines, 73and projective geometry, 73 for geodesics, 10, 41, 43

linear fractional transformation, 115, 117

as isometry, 116 as product of inversions, 133 complex, 131 fixed points, 124 multiplier, 124 Poincare interpretation, 117 preserves angles, 124 preserves circles, 125 preserves cross-ratio, 124 real, 116, 123, 125

linear point series, 74 linear transformations, 48, 75

and noneuclidean geometry, 137 two kinds, 76

Liouville metric, 3, 35, 38, 53 Lipschitz, 73 Lobachevsky, 2, 38

angle of parallelism formula,17

doctrine of, 2, 7 horocycle formula, 26 Theory of Parallels, 16-18, 21,

26, 58loxodromic transformation, 132

mapping flexible surfaces, 9 measure

additivity, 74, 78and foundations of geometry,

65angle, 63 Cayley, 69, 72 curvature of, 63, 86 euclidean, 65, 85

Index 151

general projective, 76 higher-dimensional, 88 length, 63motion invariance, 74, 93 of curvature, 86 on curved surface, 72 on linear point series, 74 on planar pencil, 74 parabolic, 85 projective, 69 space, 107special projective, 76 standard euclidean, 65 tangential, 85

Milnor, 5, 35 Minding, 2, 18, 57 minimal surface, 2 Mobius, 48 model

conformal, 36, 38 conformal disc, 120 half-space, 38 hemisphere, 35, 38 hyperboloid, 120 Klein disc, 35 Poincare disc, 35 Poincare half-plane, 35 projective, 3

modularfunction, 125 group, 119

as isometry group, 119 tessellation, 119

modulus, 119 motion, 91

as rotation, 93 of hyperbolic plane, 64 of hyperbolic space, 117

multiplier, 124noneuclidean

planimetry, 26 stereometry, 26

noneuclidean circle semiperimeter, 16

noneuclidean geometry, 58and linear transformations, 137 and quadratic forms, 121 in pseudospherical geometry, 15 named by Gauss, 71

orthogonalaxes, 10, 46, 48 surfaces, 28 trajectories, 22, 47, 54 transformation, 48, 50

parabolicgeometry, 99 involution, 72 point, 72 rotation, 64transformation, 125, 131

parallelaxiom, 64, 111 curve, 23geodesics, 14, 15, 57

parallels, 91 pencil, 74 periodicity, 114

double, 114 plane geometry, 9

euclidean, 8 planimetry, 8

noneuclidean, 9, 13, 16, 26, 58 Pliicker, 63 Poincare, 4, 113

definition of area, 129, 137 definition of congruence, 127,

136definition of length, 115, 116,

128, 136definition of volume, 137 disc model, 35 enters the omnibus, 113 half-plane model, 35

152 Index

interpretation of linear fractional transformations, 117

pointat infinity, 14, 44, 47, 54, 57,

63, 72, 76in the three geometries, 72

hyperbolic, 21 ideal, 22

polarreciprocity, 93 transformation, 50

postulateof the line, 9 straight line, 8

projectioncentral, 3, 29 perpendicular, 36 stereographic, 36, 51, 120

projectivegeometry, 63 measure, 69 model, 3 plane, 63transformation, 3, 4

projective geometry, 4and linear equations, 73 in constant curvature spaces,

73independence of measure, 109

pseudogeometry, 142 pseudosphere, 1, 25, 38

not so-called by Beltrami, 3 trigonometry on, 2 universal cover, 2

pseudospherical geometry, 58 pseudospherical surface, 3, 13 Pythagoras’ theorem, 118

quadratic forms, 118and noneuclidean geometry, 121 binary, 118 definite ternary, 139 equivalent, 118

indefinite ternary, 139 Lagrange memoir, 114 reduced, 119 ternary, 113, 120

real transformation, 125 reflection

hyperbolic, 117 in a circle, 117 in a line, 117

Riemann, 4, 69constant curvature metric, 36,

51curvature definition, 52 differential geometry, 35 essay, 35, 41, 71 finite space, 71 flatness definition, 54 space of positive curvature, 61

rotation, 48, 93about ideal centre, 98 elliptic, 64 hyperbolic, 64 parabolic, 64

scale, 77construction by motion, 75 of equidistant elements, 75 of points on a line, 65 subdivision of, 76, 77

Schering, 108 Schwarz, 115

tessellation, 115 Selling, 139 similarity

in parabolic geometry, 101 substitutions, 120

simply connected, 11, 44 but finite, 61 surface, 2, 3

spaceeuclidean, 27 hyperbolic, 38

Index 153

sphere, 29geodesic, 59 in hyperbolic space, 38 n-dimensional, 38 of imaginary radius, 99 parallel curve on, 23

sphericalgeometry, 9

in pseudospherical geometry, 62

triangle, 72 formulae, 18

trigonometry, 18stereographic projection, 36, 51, 120 stereometry

noneuclidean, 26, 59 straight line, 8

determined by two points, 8 postulate, 8, 9

superimposability, 8, 47 by rotation, 48

superposition, 9, 16of horocycles, 2, 4, 26 of horospheres, 54 of pseudospherical surface, 33 principle of, 58

surfaceconstant curvature, 8 minimal, 2 of revolution, 23, 25 orthogonal, 28 pseudospherical, 13 universal covering, 2 wrapping, 24, 25

tangency of measures, 85 ternary quadratic forms, 113, 120

definite, 139 indefinite, 139 similarity substitutions, 120

tessellationeuclidean, 114 modular, 119

of hyperbolic space, 118 of hyperboloid, 121 Schwarz, 115 symmetry group, 118

Thurston, 38 tractrix, 1, 25 transformation

complex, 131 elliptic, 125, 132 group, 65, 92, 124 hyperbolic, 126, 132 linear fractional, 115, 116 loxodromic, 132 of quadratic form, 118 parabolic, 125, 131 real, 125real linear fractional, 116

translation, 66, 93 triangle

angle sum, 18 geodesic, 15, 18, 19, 55 spherical, 18, 72

trigonometryon horosphere, 60 spherical, 18, 62

universal cover, 2 of catenoid, 2 of pseudosphere, 2

volumePoincare definition, 137

von Staudt, 109

Wachter, 38 wrapping

surface of revolution, 24, 25

Sources

Geom etryn Sti

This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincare that brought hyperbolic geometry into the mainstream of mathematics. A recognition of Beltrami comparable to that given the pioneering works of Bolyai and Lobachevsky seems long overdue— not only because Beltrami rescued hyperbolic geometry from oblivion by proving it to be logically consistent, but because he gave it a concrete meaning (a model) that made hyperbolic geometry part of ordinary mathematics.

The models subsequently discovered by Klein and Poincare brought hyperbolic geometry even further down to earth and paved the way for the current explosion of activity in low-dimensionalgeometry and topology.

By placing the works of these three mathematicians side by side and providing commentaries, this book gives the student, historian, or professional geometer a bird's-eye view of one of the great episodes in mathematics.The unified setting and historical context reveal the insights of Beltrami, Klein, and Poincare in their full brilliance.

ISBN 0-8218-0922-9

A M S on the W e bw w w .a m s .o r g

9780821809228