# Non-Euclidean Geometry: Sixth Edition

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*H. S. M. Coxeter*

MAA Press: An Imprint of the American Mathematical Society

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
continuity, with no mention of the measurement of distances or
angles. This synthetic development is followed by the introduction of
homogeneous coordinates, beginning 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
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 (in two dimensions), planes (in
three dimensions), and vice versa.

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
clarifies some obscurities in the fifth, and includes a new section
15.9 on the author's useful concept of inversive distance.

#### Reviews & Endorsements

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

No living geometer writes more clearly and beautifully about difficult topics than world famous professor H. S. M. Coxeter. When non-Euclidean geometry was first developed, it seemed little more than a curiosity with no relevance to the real world. Then to everyone's amazement, it turned out to be essential to Einstein's general theory of relativity! Coxeter's book has remained out of print for too long. Hats off to the MAA for making this classic available once more.

-- Martin Gardner

Coxeter's geometry books are a treasure that should not be lost. I am delighted to see “Non-Euclidean Geometry” back in print.

-- Doris Schattschneider

# Table of Contents

## Non-Euclidean Geometry: Sixth Edition

Table of Contents pages: 1 2

- Front Cover Cover11
- NON-EUCLIDEAN GEOMETRY iii4
- Copyright Page ii3
- PREFACE TO THE SIXTH EDITION ix10
- CONTENTS xiii14
- CHAPTER I. THE HISTORICAL DEVELOPMENT OF NON-EUCLIDEAN GEOMETRY 120
- CHAPTER II. REAL PROJECTIVE GEOMETRY: FOUNDATIONS 1635
- CHAPTER III. REAL PROJECTIVE GEOMETRY: POLARITIES, CONICS AND QUADRICS 4867
- CHAPTER IV. HOMOGENEOUS COORDINATES 7190
- 4.1 The von Staudt-Hessenberg calculus of points 7190
- 4.2 One-dimensional projectivities 7493
- 4.3 Coordinates in one and two dimensions 7695
- 4.4 Collineations and coordinate transformations 81100
- 4.5 Polarities 85104
- 4.6 Coordinates in three dimensions 87106
- 4.7 Three-dimensional projectivities 90109
- 4.8 Line coordinates for the generators of a quadric 93112
- 4.9 Complex projective geometry 94113

- CHAPTER V. ELLIPTIC GEOMETRY IN ONE DIMENSION 95114
- CHAPTER VI. ELLIPTIC GEOMETRY IN TWO DIMENSIONS 109128
- 6.1 Spherical and elliptic geometry 109128
- 6.2 Reflection 110129
- 6.3 Rotations and angles 111130
- 6.4 Congruence 113132
- 6.5 Circles 115134
- 6.6 Composition of rotations 118137
- 6.7 Formulae for distance and angle 120139
- 6.8 Rotations and quaternions 122141
- 6.9 Alternative treatment using the complex plane 126145

- CHAPTER VII. ELLIPTIC GEOMETRY IN THREE DIMENSIONS 128147
- 7.1 Congruent transformations 128147
- 7.2 Clifford parallels 133152
- 7.3 The Stephanos-Cartan representation of rotations by points 136155
- 7.4 Right translations and left translations 138157
- 7.5 Right parallels and left parallels 141160
- 7.6 Study's representation of lines by pairs of points 146165
- 7.7 Clifford translations and quaternions 148167
- 7.8 Study's coordinates for a line 151170
- 7.9 Complex space 153172

- CHAPTER VIII. DESCRIPTIVE GEOMETRY 157176
- 8.1 Klein's projective model for hyperbolic geometry 157176
- 8.2 Geometry in a convex region 159178
- 8.3 Veblen's axioms of order 161180
- 8.4 Order in a pencil 162181
- 8.5 The geometry of lines and planes through a fixed point 164183
- 8.6 Generalized bundles and pencils 165184
- 8.7 Ideal points and lines 171190
- 8.8 Verifying the projective axioms 172191
- 8.9 Parallelism 174193

- CHAPTER IX EUCLIDEAN AND HYPERBOLIC GEOMETRY 179198
- CHAPTER X. HYPERBOLIC GEOMETRY IN TWO DIMENSIONS 199218
- 10.1 Ideal elements 199218
- 10.2 Angle-bisectors 200219
- 10.3 Congruent transformations 201220
- 10.4 Some famous constructions 204223
- 10.5 An alternative expression for distance 206225
- 10.6 The angle of parallelism 207226
- 10.7 Distance and angle in terms of poles and polars 208227
- 10.8 Canonical coordinates 209228
- 10.9 Euclidean geometry as a limiting case 211230

- CHAPTER XI. CIRCLES AND TRIANGLES 213232
- CHAPTER XII. THE USE OF A GENERAL TRIANGLE OF REFERENCE 224243
- CHAPTER XIII. AREA 241260
- 13.1 Equivalent regions 241260
- 13.2 The choice of a unit 241260
- 13.3 The area of a triangle in elliptic geometry 242261
- 13.4 Area in hyperbolic geometry 243262
- 13.5 The extension to three dimensions 247266
- 13.6 The differential of distance 248267
- 13.7 Arcs and areas of circles 249268
- 13.8 Two surfaces which can be developed on the Euclidean plane 251270

- CHAPTER XIV. EUCLIDEAN MODELS 252271
- 14.1 The meaning of "elliptic" and "hyperbolic" 252271
- 14.2 Beltrami's model 252271
- 14.3 The differential of distance 254273
- 14.4 Gnomonic projection 255274
- 14.5 Development on surfaces of constant curvature 256275
- 14.6 Klein's conformal model of the elliptic plane 258277
- 14.7 Klein's conformal model of the hyperbolic plane 260279
- 14.8 Poincaré's model of the hyperbolic plane 263282
- 14.9 Conformal models of non-Euclidean space 264283

- CHAPTER XV. CONCLUDING REMARKS 267286
- 15.1 Hjelmslev's mid-line 267286
- 15.2 The Napier chain 273292
- 15.3 The Engel chain 277296
- 15.4 Normalized canonical coordinates 281300
- 15.5 Curvature 283302
- 15.6 Quadratic forms 284303
- 15.7 The volume of a tetrahedron 285304
- 15.8 A brief historical survey of construction problems 289308
- 15.9 Inversive distance and the angle of parallelism 292311

- Appendix: Angles and Arcs in the Hyperbolic Plane 299318
- Bibliography 317336
- Index 327346
- Back Cover 337356

- Front Cover Cover1357
- NON-EUCLIDEAN GEOMETRY iii360
- Copyright Page ii359
- PREFACE TO THE SIXTH EDITION ix366
- CONTENTS xiii370
- CHAPTER I. THE HISTORICAL DEVELOPMENT OF NON-EUCLIDEAN GEOMETRY 1376
- CHAPTER II. REAL PROJECTIVE GEOMETRY: FOUNDATIONS 16391
- CHAPTER III. REAL PROJECTIVE GEOMETRY: POLARITIES, CONICS AND QUADRICS 48423
- CHAPTER IV. HOMOGENEOUS COORDINATES 71446
- 4.1 The von Staudt-Hessenberg calculus of points 71446
- 4.2 One-dimensional projectivities 74449
- 4.3 Coordinates in one and two dimensions 76451
- 4.4 Collineations and coordinate transformations 81456
- 4.5 Polarities 85460
- 4.6 Coordinates in three dimensions 87462
- 4.7 Three-dimensional projectivities 90465
- 4.8 Line coordinates for the generators of a quadric 93468
- 4.9 Complex projective geometry 94469

- CHAPTER V. ELLIPTIC GEOMETRY IN ONE DIMENSION 95470

Table of Contents pages: 1 2