Table of Contents
Introduction 8
Chapter 1 History of Geometry 15
Ancient Geometry: Practical and Empirical 15
Finding the Right Angle 18
Locating the Inaccessible 19
Estimating the Wealth 19
Ancient Geometry: Abstract and Applied 21
Doubling the Cube 22
Trisecting the Angle 23
Squaring the Circle 24
Idealization and Proof 25
The Euclidean Synthesis 27
Gnomonics and the Cone 28
Astronomy and Trigonometry 30
Ancient Geometry: Cosmological and Metaphysical 31
Pythagorean Numbers and Platonic Solids 31
Measuring the Earth And Heavens 34
The Post-Classical Period 36
Passage Through Islam 36
Europe Rediscovers the Classics 38
Linear Perspective 40
Transformation 41
French Circles 41
Geometrical Calculus 45
The World System 48
Relaxation and Rigour 50
Protection Again 51
Non-Euclidean Geometries 52
A Grand Synthesis 56
The Real World 59
Chapter 2 Ranches of Geometry 61
Euclidean Geometry 61
Fundamentals 62
Plane Geometry 64
Pythagorean Theorem 68
Circles 68
Regular Polygons 70
Conic Sections and Geometric Art 71
Solid Geometry 72
Volume 73
Regular Solids 73
Conic Section 74
Analytic Geometry 78
Elementary Analytic Geometry 79
Analytic Geometry of Three and More Dimensions 84
Algebraic Geometry 87
Projective Geometry 89
Parallel Lines and the Projection of Infinity 91
Projective Invariants 92
Projective Conic Sections 96
Differential Geometry 98
Curvature of Curves 101
Curvature of Surfaces 103
Shortest Paths on a Surface 106
Non-Euclidean Geometry 108
Spherical Geometry 110
Hyperbolic Geometry 112
Topology 115
Simply Connected 115
Topological Equivalence 117
Homeomorphism 119
Topological Structure 120
Topological Space 120
Continuity 122
Algebraic Topology 122
Knot Theory 125
Graph Theory 128
Chapter 3 Geometric Terms and Concepts 139
Algebraic Surface 139
Angle Trisection: Archimedes' Method 140
Trisecting The Angle: The Quadratrix of Hippias 141
Axiom 142
Axiomatic Method 143
Brachistochrone 144
Bridge of Asses 144
Brouwer's Fixed Point Theorem 146
Catenary 147
Ceva's Theorem 149
Circle 149
Compactness 150
Cone 151
Coordinate Systems 152
Cross Ratio 153
Curve 154
Cycloid 155
Cylinder 156
Desargues's Theorem 157
Dimension 158
Duality 159
The Elements Since The Middle Ages 161
Ellipse 163
Ellipsoid 164
Envelope 165
Euclid's Windmill 166
Euclidean Space 168
Method of Exhaustion 169
Fractal 170
Golden Ratio 172
Graph 174
Harmonic Construction 176
Hausdorff Space 177
Hilbert Space 179
Hippocrates' Quadrature of the Lone 180
Hyperbola 182
Hyperboloid 183
Incommensurables 184
Isometric Drawing 186
Konigsberg Bridge Problem 187
Line 189
Measuring the Earth, Classical and Arabic 189
Measuring the Earth, Modern 191
Metric Space 192
Parabola 194
Paraboloid 195
Parallel Postulate 196
Pencil 196
Pi 197
Platonic Solid 198
Polygon 200
Projection 200
Pythagorean Theorem 201
Space-Time 204
Spiral 207
Square 209
Thales' Rectangle 209
Topological Space 211
Chapter 4 Biographies of Great Geometers 213
Ancient Greek and Islamic Geometers 214
Apollonius of Perga 214
Archimedes 218
Archytas of Tarentum 226
Conon of Samos 228
Eratosthenes of Cyrene 229
Euclid 231
Renditions of the Elements 236
Eudoxus of Cnidus 238
Heron of Alexandria 241
Hippias of Elis 244
Hippocrates of Chios 244
Menaechmus 246
Omar Khayyam 247
Pappus of Alexandria 250
Pythagoras 253
Thales of Miletus 255
Theaetetus 258
Pre-Modern (Pre-1800) Geometers 259
Bonaventura Cavalleri 259
Giovanni Ceva 260
Girard Desargues 262
René Descartes 265
Leonhard Euler 266
Gaspard Monge, Count De Péluse 270
Gilles Personne De Roberval 274
Simon Stevin 275
Modern Geometers 277
Lars Valerian Ahlfors 277
Pavel Sergeevich Aleksandrov 278
James W. Alexander II 280
Sir Michael Francis Atiyah 281
Eugenio Beltrami 284
Enrico Betti 285
János Bolyai 287
Charles-Julien Brianchon 288
Luitzen Egbertus Jan Brouwer 289
Michel Chasles 291
Shiing-Shen Chern 293
William Kingdon Clifford 294
Pierre René Deligne 296
Simon Kirwan Donaldson 297
Vladimir Gershonovich Drinfeld 298
Alexandre Grothendieck 299
David Hilbert 301
Gaston Maurice Julia 305
Felix Klein 306
Niels Fabian Helge von Koch 308
Kodaira Kunihiko 309
Nikolay Ivanovich Lobachevsky 311
Benoit Mandelbrot 313
John Willard Milnor 315
Hermann Minkowski 316
August Ferdinand Möbius 318
Mori Shigefumi 319
David Bryant Mumford 320
Sergey Petrovich Novikov 321
Grigori Perelman 323
Henri Poincaré 324
Jean-Victor Poncelet 331
Bernhard Riemann 332
Jean-Pierre Serre 337
Waclaw Sierpinski 339
Stephen Smale 341
Karl Georg Christian von Staudt 343
Jakob Steiner 344
René Frédéric Thom 345
William Paul Thurston 347
Oswald Veblen 348
Vladimir Voevodsky 350
André Weil 351
Wendelin Werner 354
Shing-Tung Yau 355
Glossary 357
Bibliography 360
Index 364
