history of mathematics volume 10 sources of hyperbolic · geometry, hyperbolic—history—sources....
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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
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