The Britannica Guide to Geometry

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ISBN: 9781615301027 | 161530102X
Cover: Hardcover
Copyright: 8/15/2010

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More than a study of shapes and angles, geometry reflects an amalgamation of discoveries over time. This book not only provides readers with a comprehensive understanding of geometric shapes, axioms, and formulas, it presents the fields brilliant mindsfrom Euclid to Wendelin Werner and many in betweenwhose works reflect a progression of mathematical thought throughout the centuries and have helped produce the various branches of geometry as they are known today. Detailed diagrams illustrate various concepts and help make geometry accessible to all.

Introduction	p. 12
History of Geometry	p. 21
Ancient Geometry: Practical and Empirical	p. 22
Finding the Right Angle	p. 23
Locating the Inaccessible	p. 24
Estimating the Wealth	p. 25
Ancient Geometry: Abstract and Applied	p. 26
The Three Classical Problems	p. 26
Doubling the Cube	p. 27
Trisecting the Angle	p. 27
Squaring the Circle	p. 28
Idealization and Proof	p. 29
The Euclidean Synthesis	p. 30
Gnomonics and the Cone	p. 32
Astronomy and Trigonometry	p. 33
Calculation	p. 33
Epistemology	p. 34
Ancient Geometry: Cosmological and Metaphysical	p. 34
Pythagorean Numbers and Platonic Solids	p. 34
Measuring the Earth and Heavens	p. 36
The Post-Classical Period	p. 38
Passage Through Islam	p. 38
Europe Rediscovers the Classics	p. 40
Linear Perspective	p. 41
Transformation	p. 43
French Circles	p. 43
Projective Geometry	p. 43
Cartesian Geometry	p. 45
Geometrical Calculus	p. 46
The World System	p. 49
Relaxation and Rigour	p. 50
Projection Again	p. 51
Non-Euclidean Geometries	p. 52
A Grand Synthesis	p. 54
The Real World	p. 57
Branches of Geometry	p. 59
Euclidean Geometry	p. 59
Fundamentals	p. 60
Plane Geometry	p. 61
Congruence of Triangles	p. 61
Similarity of Triangles	p. 62
Areas	p. 63
Pythagorean Theorem	p. 64
Circles	p. 65
Regular Polygons	p. 66
Conic Sections and Geometric Art	p. 67
Solid Geometry	p. 67
Volume	p. 68
Regular Solids	p. 68
Conic Section	p. 69
Analytic Definition	p. 70
Greek Origins	p. 70
Post-Greek Applications	p. 71
Analytic Geometry	p. 73
Elementary Analytic Geometry	p. 73
Analytic Geometry of Three and More Dimensions	p. 77
Vector Analysis	p. 78
Projections	p. 79
Algebraic Geometry	p. 79
Projective Geometry	p. 81
Parallel Lines and the Projection of Infinity	p. 82
Projective Invariants	p. 83
Projective Conic Sections	p. 84
Differential Geometry	p. 85
Curvature of Curves	p. 87
Curvature of Surfaces	p. 89
Shortest Paths on a Surface	p. 91
Non-Euclidean Geometry	p. 92
Comparison of Euclidean, Spherical, and Hyperbolic Geometries	p. 93
Spherical Geometry	p. 94
Hyperbolic Geometry	p. 96
Topology	p. 97
Basic Concepts of General Topology	p. 98
Simply Connected	p. 98
Topological Equivalence	p. 99
Homeomorphism	p. 100
Topological Structure	p. 101
Algebraic Topology	p. 103
Fundamental Group	p. 104
Knot Theory	p. 105
Graph Theory	p. 108
Biographies	p. 115
Ancient Greek and Islamic Geometers	p. 116
Apollonius of Perga	p. 116
Archimedes	p. 119
His Life	p. 119
His Works	p. 120
His Influence	p. 124
Archytas of Tarentum	p. 126
Conon of Samos	p. 127
Eratosthenes of Cyrene	p. 128
Euclid	p. 129
Life	p. 129
Sources and Contents of the Elements	p. 130
Euclid's Axioms and Euclid's Common Notions	p. 131
Renditions of the Elements	p. 134
Other Writings	p. 136
Assessment	p. 136
Eudoxus of Cnidus	p. 137
Life	p. 137
Mathematician	p. 138
Assessment	p. 139
Heron of Alexandria	p. 139
Hippias of Elis	p. 142
Hippocrates of Chios	p. 142
Menaechmus	p. 143
Omar Khayyam	p. 144
Pappus of Alexandria	p. 147
Pythagoras	p. 149
Thales of Miletus	p. 151
Theaetetus	p. 153
Pre-Modern (Pre-1800) Geometers	p. 154
Bonaventura Cavalieri	p. 154
Giovanni Ceva	p. 155
Girard Desargues	p. 156
René Descartes	p. 158
Leonhard Euler	p. 160
Gaspard Monge, count de Péluse	p. 163
Gilles Personne de Roberval	p. 167
Simon Stevin	p. 168
Modern Geometers	p. 169
Lars Valerian Ahlfors	p. 169
Pavel Sergeevich Aleksandrov	p. 170
James W Alexander II	p. 172
Sir Michael Francis Atiyah	p. 173
Eugenio Beltrami	p. 174
Enrico Betti	p. 175
János Bolyai	p. 176
Charles-Julien Brianchon	p. 177
Luitzen Egbertus Jan Brouwer	p. 178
Michel Chasles	p. 180
Shiing-shen Chern	p. 181
William Kingdon Clifford	p. 182
Pierre René Deligne	p. 184
Simon Kirwan Donaldson	p. 185
Vladimir Gershonovich Drinfeld	p. 186
Alexandre Grothendieck	p. 187
David Hilbert	p. 188
Gaston Maurice Julia	p. 192
Felix Klein	p. 193
Niels Fabian Helge von Koch	p. 195
Kodaira Kunihiko	p. 197
Nikolay Ivanovich Lobachevsky	p. 198
Benoit Mandelbrot	p. 200
John Willard Milnor	p. 202
Hermann Minkowski	p. 203
August Ferdinand Möbius	p. 204
Mori Shigefumi	p. 206
David Bryant Mumford	p. 207
Sergey Petrovich Novikov	p. 208
Grigori Perelman	p. 209
Henri Poincaré	p. 210
Jean-Victor Poncelet	p. 216
Bernhard Riemann	p. 217
Jean-Pierre Serre	p. 221
Waclaw Sierpinski	p. 223
Stephen Smale	p. 225
Karl Georg Christian von Staudt	p. 226
Jakob Steiner	p. 227
René Frédéric Thom	p. 228
William Paul Thurston	p. 229
Oswald Veblen	p. 231
Vladimir Voevodsky	p. 233
André Weil	p. 234
Wendelin Werner	p. 236
Shing-Tung Yau	p. 237
Appendix of Geometric Terms and Concepts	p. 238
Algebraic Surface	p. 238
Angle Trisection: Archimedes' Method	p. 240
Angle Trisection: Quadratrix of Hippias	p. 241
Axiom	p. 241
Axiomatic Method	p. 242
Brachistochrone	p. 243
Bridge of Asses	p. 244
Brouwer's Fixed Point Theorem	p. 245
Catenary	p. 246
Ceva's Theorem	p. 248
Circle	p. 248
Compactness	p. 249
Cone	p. 250
Coordinate Systems	p. 251
Cross Ratio	p. 252
Curve	p. 253
Cycloid	p. 253
Cylinder	p. 254
Desargues's Theorem	p. 255
Dimension	p. 256
Duality	p. 257
The Elements Since the Middle Ages	p. 259
Ellipse	p. 261
Ellipsoid	p. 262
Envelope	p. 263
Euclid's Windmill	p. 263
Euclidean Space	p. 265
Method of Exhaustion	p. 266
Fractal	p. 266
Golden Ratio	p. 268
Graph	p. 270
Harmonic Construction	p. 272
Hausdorff Space	p. 273
Hilbert Space	p. 274
Hippocrates' Quadrature of the Lune	p. 276
Hyperbola	p. 277
Hyperboloid	p. 278
Incommensurables	p. 279
Isometric Drawing	p. 280
Königsberg Bridge Problem	p. 281
Line	p. 283
Measuring the Earth, Classical and Arabic	p. 283
Measuring the Earth, Modern	p. 284
Metric Space	p. 286
Parabola	p. 287
Paraboloid	p. 288
Parallel Postulate	p. 289
Pencil	p. 289
Pi	p. 290
Platonic Solid	p. 291
Polygon	p. 292
Projection	p. 292
Pythagorean Theorem	p. 293
Space-Time	p. 296
Spiral	p. 298
Square	p. 300
Thales' Rectangle	p. 301
Topological Space	p. 302
Glossary	p. 303
Bibliography	p. 306
Index	p. 310
Table of Contents provided by Ingram. All Rights Reserved.

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