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i i i For use by the Norwegian University of Science and Technology i The Foundations of Geometry Second Edition Gerard A. Venema Department of Mathematics and Statistics Calvin College Upper Saddle River, New Jersey 07458 Copyright Pearson. All Rights Reserved. i i i i i i i For use by the Norwegian University of Science and Technology i Library of Congress Cataloging-in-Publication Data Venema, Gerard. The foundations of geometry / Gerard A. Venema. -- 2nd ed. p. cm. Includes bibliographical references index. ISBN 0-13-602058-5 1. Geometry--Foundations--Textbooks. I. Title. QA445.V455 2012 2012 514--dc22 2011002700 CIP Acquisitions Editor: Marnie Greenhut Assistant Editor: Elle Driska Senior Managing Editor: Karen Wernholm Production Project Manager: Beth Houston Marketing Manager: Roxanne McCarley Marketing Assistant: Caitlin Crane Senior Author Support/Technology Specialist: Joe Vetere Senior Manufacturing Buyer: Carol Melville Design Manager: Andrea Nix Senior Design Specialist: Heather Scott Cover Image Credit: Science and Society/Superstock 2012, 2006 Pearson Education, Inc. Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Pearson Prentice HallTM is a trademark of Pearson Education, Inc. Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN 0-13-602058-5 Pearson Education LTD., London Pearson Education Australia PTY, Limited, Sydney Pearson Education Singapore, Pte. Ltd Pearson Education North Asia Ltd, Hong Kong Pearson Education Canada, Ltd., Toronto Pearson Educacion de Mexico, S.A. de C.V. Pearson Education - Japan, Tokyo Pearson Education Malaysia, Pte. Ltd Copyright Pearson. All Rights Reserved. i i i i i i i For use by the Norwegian University of Science and Technology i To my family: Patricia, Sara, Emily, Daniel, David, Christian, and Ian. Copyright Pearson. All Rights Reserved. i i i i i i i For use by the Norwegian University of Science and Technology i Copyright Pearson. All Rights Reserved. i i i i i i i For use by the Norwegian University of Science and Technology i Contents Preface ix 1 Prologue: Euclid’s Elements 1 1.1 Geometry before Euclid . 1 1.2 The logical structure of Euclid’s Elements ................... 2 1.3 The historical significance of Euclid’s Elements ................ 3 1.4 A look at Book I of the Elements ........................ 4 1.5 A critique of Euclid’s Elements ......................... 7 1.6 Final observations about the Elements ..................... 9 2 Axiomatic Systems and Incidence Geometry 14 2.1 The structure of an axiomatic system . 14 2.2 An example: Incidence geometry . 16 2.3 The parallel postulates in incidence geometry . 20 2.4 Axiomatic systems and the real world . 22 2.5 Theorems, proofs, and logic . 24 2.6 Some theorems from incidence geometry . 32 3 Axioms for Plane Geometry 35 3.1 The undefined terms and two fundamental axioms . 36 3.2 Distance and the Ruler Postulate . 37 3.3 Plane separation . 46 3.4 Angle measure and the Protractor Postulate . 51 3.5 The Crossbar Theorem and the Linear Pair Theorem . 55 3.6 The Side-Angle-Side Postulate . 62 3.7 The parallel postulates and models . 66 4 Neutral Geometry 69 4.1 The Exterior Angle Theorem and existence of perpendiculars . 70 4.2 Triangle congruence conditions . 74 4.3 Three inequalities for triangles . 77 4.4 The Alternate Interior Angles Theorem . 82 4.5 The Saccheri-Legendre Theorem . 84 4.6 Quadrilaterals . 88 4.7 Statements equivalent to the Euclidean Parallel Postulate . 91 4.8 Rectangles and defect . 97 4.9 The Universal Hyperbolic Theorem . 104 5 Euclidean Geometry 106 5.1 Basic theorems of Euclidean geometry . 107 5.2 The Parallel Projection Theorem . 110 5.3 Similar triangles . 112 5.4 The Pythagorean Theorem . 114 5.5 Trigonometry . 116 5.6 Exploring the Euclidean geometry of the triangle . 118 v Copyright Pearson. All Rights Reserved. i i i i i i i For use by the Norwegian University of Science and Technology i vi Contents 6 Hyperbolic Geometry 131 6.1 Basic theorems of hyperbolic geometry . 133 6.2 Common perpendiculars . 138 6.3 The angle of parallelism . 140 6.4 Limiting parallel rays . 144 6.5 Asymptotic triangles . 149 6.6 The classification of parallels . 152 6.7 Properties of the critical function . 155 6.8 The defect of a triangle . 160 6.9 Is the real world hyperbolic? . 164 7 Area 167 7.1 The Neutral Area Postulate . 167 7.2 Area in Euclidean geometry . 173 7.3 Dissection theory . 180 7.4 Proof of the dissection theorem in Euclidean geometry . 181 7.5 The associated Saccheri quadrilateral . 185 7.6 Area and defect in hyperbolic geometry . 190 8 Circles 195 8.1 Circles and lines in neutral geometry . 195 8.2 Circles and triangles in neutral geometry . 200 8.3 Circles in Euclidean geometry . 206 8.4 Circular continuity . 212 8.5 Circumference and area of Euclidean circles . 215 8.6 Exploring Euclidean circles . 222 9 Constructions 228 9.1 Compass and straightedge constructions . 228 9.2 Neutral constructions . 231 9.3 Euclidean constructions . 234 9.4 Construction of regular polygons . 235 9.5 Area constructions . 239 9.6 Three impossible constructions . 242 10 Transformations 245 10.1 Properties of isometries . 246 10.2 Rotations, translations, and glide reflections . 252 10.3 Classification of Euclidean motions . 260 10.4 Classification of hyperbolic motions . 263 10.5 A transformational approach to the foundations . 265 10.6 Similarity transformations in Euclidean geometry . 270 10.7 Euclidean inversions in circles . 275 11 Models 287 11.1 The Cartesian model for Euclidean geometry . 289 11.2 The Poincare´ disk model for hyperbolic geometry . 290 11.3 Other models for hyperbolic geometry . 296 11.4 Models for elliptic geometry . 301 12 Polygonal Models and the Geometry of Space 303 12.1 Curved surfaces . 304 Copyright Pearson. All Rights Reserved. i i i i i i i For use by the Norwegian University of Science and Technology i Contents vii 12.2 Approximate models for the hyperbolic plane . 315 12.3 Geometric surfaces . 321 12.4 The geometry of the universe . 326 12.5 Conclusion . 331 12.6 Further study . 331 12.7 Templates . 336 APPENDICES A Euclid’s Book I 344 A.1 Definitions . 344 A.2 Postulates . 346 A.3 Common Notions . 346 A.4 Propositions . 346 B Systems of Axioms for Geometry 350 B.1 Hilbert’s axioms . 351 B.2 Birkhoff’s axioms . 353 B.3 MacLane’s axioms . 354 B.4 SMSG axioms . 355 B.5 UCSMP axioms . 357 C The Postulates Used in this Book 360 C.1 Criteria used in selecting the postulates . 360 C.2 Statements of the postulates . 362 C.3 Logical relationships . 363 D The van Hiele Model 365 E Set Notation and the Real Numbers 366 E.1 Some elementary set theory . 366 E.2 Axioms for the real numbers . 367 E.3 Properties of the real numbers . 368 E.4 One-to-one and onto functions . 370 E.5 Continuous functions . 371 F Hints for Selected Exercises 372 Bibliography 379 Copyright Pearson. All Rights Reserved. i i i i i i i For use by the Norwegian University of Science and Technology i Copyright Pearson. All Rights Reserved. i i i i i i i For use by the Norwegian University of Science and Technology i Preface This is a textbook for an undergraduate course in axiomatic geometry. The text is targeted at mathematics students who have completed the calculus sequence and perhaps a first course in linear algebra, but who have not yet encountered such upper-level mathematics courses as real analysis and abstract algebra. A course based on this book will enrich the education of all mathematics majors and will ease their transition into more advanced mathematics courses. The book also includes emphases that make it especially appropriate as the textbook for a geometry course taken by future high school mathematics teachers. WHAT’S NEW IN THE SECOND EDITION? For the benefit of those who have used the first edition of this book, here is a quick summary of what has changed.
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