The Britannica guide to geometry PDF
Preview The Britannica guide to geometry
Published in 2011 by Britannica Educational Publishing (a trademark of Encyclopædia Britannica, Inc.) in association with Rosen Educational Services, LLC 29 East 21st Street, New York, NY 10010. Copyright © 2011 Encyclopædia Britannica, Inc. Britannica, Encyclopædia Britannica, and the Thistle logo are registered trademarks of Encyclopædia Britannica, Inc. All rights reserved. Rosen Educational Services materials copyright © 2010 Rosen Educational Services, LLC. All rights reserved. Distributed exclusively by Rosen Educational Services. For a listing of additional Britannica Educational Publishing titles, call toll free (800) 237-9932. First Edition Britannica Educational Publishing Michael I. Levy: Executive Editor J.E. Luebering: Senior Manager Marilyn L. Barton: Senior Coordinator, Production Control Steven Bosco: Director, Editorial Technologies Lisa S. Braucher: Senior Producer and Data Editor Yvette Charboneau: Senior Copy Editor Kathy Nakamura: Manager, Media Acquisition William L. Hosch: Associate Editor, Mathematics and Computer Sciences Rosen Educational Services Hope Lourie Killcoyne: Senior Editor and Project Manager Joanne Randolph: Editor Nelson Sá: Art Director Matthew Cauli: Designer, Cover Design Cindy Reiman: Photography Manager Introduction by John Strazzabosco Library of Congress Cataloging-in-Publication Data The Britannica guide to geometry / edited by Robert Curley.—1st ed. p. cm.—(Math explained) “In association with Britannica Educational Publishing, Rosen Educational Services.” Includes bibliographical references and index. ISBN 978-1-61530-217-8 ( eBook) 1. Geometry—History. 2. Mathematicians—Biography. I. Curley, Robert, 1955– QA443.5.B75 2010 516.009—dc22 2009044163 Cover © www.istockphoto.com/thumb; p. 12 © www.istockphoto.com/RyanFox; pp. 21, 59, 115, 238, 303, 206, 310 © Hulton Archive/Getty Images. C 21 ontents Introduction 12 Chapter 1: History of Geometry 21 Ancient Geometry: Practical and Empirical 22 Finding the Right Angle 23 Locating the Inaccessible 24 Estimating the Wealth 25 Ancient Geometry: Abstract and Applied 26 25 The Three Classical Problems 26 Doubling the Cube 27 Trisecting the Angle 27 Squaring the Circle 28 Idealization and Proof 29 The Euclidean Synthesis 30 Gnomonics and the Cone 32 Astronomy and Trigonometry 33 Calculation 33 Epistemology 34 Ancient Geometry: Cosmological and Metaphysical 34 44 Pythagorean Numbers and Platonic Solids 34 Measuring the Earth and Heavens 36 The Post-Classical Period 38 Passage Through Islam 38 Europe Rediscovers the Classics 40 Linear Perspective 41 Transformation 43 French Circles 43 Projective Geometry 43 Cartesian Geometry 45 Geometrical Calculus 46 47 The World System 49 Relaxation and Rigour 50 Projection Again 51 Non-Euclidean Geometries 52 A Grand Synthesis 54 The Real World 57 Chapter 2: Branches of Geometry 59 Euclidean Geometry 59 Fundamentals 60 Plane Geometry 61 Congruence of Triangles 61 Similarity of Triangles 62 Areas 63 Pythagorean Theorem 64 Circles 65 Regular Polygons 66 Conic Sections and Geometric Art 67 Solid Geometry 67 Volume 68 Regular Solids 68 65 Conic Section 69 Analytic Definition 70 Greek Origins 70 Post-Greek Applications 71 Analytic Geometry 73 Elementary Analytic Geometry 73 Analytic Geometry of Three and More Dimensions 77 Vector Analysis 78 Projections 79 Algebraic Geometry 79 Projective Geometry 81 72 Parallel Lines and the Projection of Infinity 82 Projective Invariants 83 87 Projective Conic Sections 84 Differential Geometry 85 Curvature of Curves 87 Curvature of Surfaces 89 Shortest Paths on a Surface 91 Non-Euclidean Geometry 92 Comparison of Euclidean, Spherical, and Hyperbolic Geometries 93 Spherical Geometry 94 Hyperbolic Geometry 96 Topology 97 Basic Concepts of General Topology 98 Simply Connected 98 Topological Equivalence 99 Homeomorphism 100 Topological Structure 101 Algebraic Topology 103 Fundamental Group 104 Knot Theory 105 Graph Theory 108 Chapter 3: Biographies 115 89 Ancient Greek and Islamic Geometers 116 120 Apollonius of Perga 116 Archimedes 119 His Life 119 His Works 120 His Influence 124 Archytas of Tarentum 126 Conon of Samos 127 Eratosthenes of Cyrene 128 Euclid 129 Life 129 Sources and Contents of the Elements 130 Euclid’s Axioms and Euclid’s 133 Common Notions 131 Renditions of the Elements 134 Other Writings 136 Assessment 136 Eudoxus of Cnidus 137 Life 137 Mathematician 138 Assessment 139 Heron of Alexandria 139 Hippias of Elis 142 Hippocrates of Chios 142 Menaechmus 143 Omar Khayyam 144 Pappus of Alexandria 147 Pythagoras 149 Thales of Miletus 151 Theaetetus 153 Pre-Modern (Pre-1800) Geometers 154 Bonaventura Cavalieri 154 Giovanni Ceva 155 Girard Desargues 156 René Descartes 158 149 Leonhard Euler 160 Gaspard Monge, count de Péluse 163 Gilles Personne de Roberval 167 Simon Stevin 168 Modern Geometers 169 Lars Valerian Ahlfors 169 Pavel Sergeevich Aleksandrov 170 James W. Alexander II 172 Sir Michael Francis Atiyah 173 Eugenio Beltrami 174 Enrico Betti 175 János Bolyai 176 158 Charles-Julien Brianchon 177 Luitzen Egbertus Jan Brouwer 178 183 Michel Chasles 180 Shiing-shen Chern 181 William Kingdon Clifford 182 Pierre René Deligne 184 Simon Kirwan Donaldson 185 Vladimir Gershonovich Drinfeld 186 Alexandre Grothendieck 187 David Hilbert 188 Gaston Maurice Julia 192 Felix Klein 193 Niels Fabian Helge von Koch 195 Kodaira Kunihiko 197 Nikolay Ivanovich Lobachevsky 198 Benoit Mandelbrot 200 John Willard Milnor 202 Hermann Minkowski 203 August Ferdinand Möbius 204 Mori Shigefumi 206 David Bryant Mumford 207 Sergey Petrovich Novikov 208 Grigori Perelman 209 Henri Poincaré 210 200 Jean-Victor Poncelet 216 Bernhard Riemann 217 Jean-Pierre Serre 221 217 Wacław Sierpi ski 223 ń Stephen Smale 225 Karl Georg Christian von Staudt 226 Jakob Steiner 227 René Frédéric Thom 228 William Paul Thurston 229 Oswald Veblen 231 Vladimir Voevodsky 233 André Weil 234 Wendelin Werner 236 Shing-Tung Yau 237
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