College Geometry PDF
Preview College Geometry
Page iii College Geometry Howard Eves University of Maine University of Central Florida Page iv Editorial, Sales, and Customer Service Offices Jones and Bartlett Publishers One Exeter Plaza Boston, MA 02116 18008320034 6178593900 Jones and Bartlett Publishers International 7 Melrose Terrace London W6 7RL England Copyright © 1995 by Jones and Bartlett Publishers, Inc. Portions of this book first appeared in Fundamentals of Modern Elementary Geometry, by Howard Eves, copyright © 1992 by Jones and Bartlett Publishers, Inc. and in A Survey of Geometry, by Howard Eves, copyright © 1972, 1963 by Allyn and Bacon, Inc. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any informational storage and retrieval system, without written permission from the copyright owner. Library of Congress CataloginginPublication Data Eves, Howard Whitley, 1911– College geometry / Howard Eves. p. cm. Includes bibliographical references (p. – ) and index. ISBN 0867204753 1. Geometry. I. Title. QA445.E88 1995 516.04dc20 9439749 CIP Printed in the United States of America 98 97 96 95 94 10 9 8 7 6 5 4 3 2 1 Page v TO DON In appreciation of those wonderful years that only identical twins can share. Page vi CONTENTS Preface ix 0 1 The Fountainhead 0.1 The Earliest Geometry 1 0.2 The Empirical Nature of PreHellenic Geometry 3 0.3 The Greek Contribution of Material Axiomatics 8 0.4 Euclid's Elements 15 0.5 The Greek Bequest 21 0.6 The Transmission of Greek Geometry to the Occident 30 0.7 The Case for Empirical, or Experimental, Geometry (Optional) 38 Bibliography 44 Part 1 47 The Fundamentals of Modern Elementary Geometry 1 49 Modern Elementary Geometry 1.1 Sensed Magnitudes 50 1.2 Infinite Elements 56 1.3 The Theorems of Menelaus and Ceva 60 1.4 Applications of the Theorems of Menelaus and Ceva 64 1.5 Cross Ratio 70 1.6 Applications of Cross Ratio 74 1.7 Homographic Ranges and Pencils (Optional) 77 1.8 Harmonic Division 81 1.9 Orthogonal Circles 86 1.10 The Radical Axis of a Pair of Circles 91 Bibliography 97 Page vii 2 99 Elementary Transformations 2.1 Transformation Theory 100 2.2 Fundamental Point Transformations of the Plane 104 2.3 Applications of the Homothety Transformation 108 2.4 Isometries 114 2.5 Similarities 118 2.6 Inversion 122 2.7 Properties of Inversion 128 2.8 Applications of Inversion 132 2.9 Reciprocation 141 2.10 Applications of Reciprocation 146 2.11 Space Transformations (Optional) 150 Bibliography 157 3 159 Euclidean Constructions 3.1 The Euclidean Tools 160 3.2 The Method of Loci 163 3.3 The Method of Transformation 167 3.4 The Double Points of Two Coaxial Homographic Ranges (Optional) 172 3.5 The MohrMascheroni Construction Theorem 175 3.6 The PonceletSteiner Construction Theorem 180 3.7 Some Other Results (Optional) 186 3.8 The Regular SeventeenSided Polygon (Optional) 192 3.9 The Impossibility of Solving the Three Famous Problems of Antiquity 200 with Euclidean Tools (Optional) Bibliography 202 Appendix A: Perspectivity Transformation (Optional) 205 Part II 215 The NonEuclidean Revolution 4 217 NonEuclidean Geometry 4.1 Historical Background 218 4.2 Parallels and Hyperparallels 226 4.3 Limit Triangles 230 4.4 Saccheri Quadrilaterals and the AngleSum of a Triangle 234 Page viii 4.5 Area of a Triangle 238 4.6 Ideal and UltraIdeal Points 246 4.7 An Application of Ideal and UltraIdeal Points 250 4.8 Geometry and Physical Space 252 Bibliography 255 5 257 The Foundations of Geometry 5.1 Some Logical Shortcomings of Euclid's Elements 258 5.2 Modern Postulational Foundations for Euclidean Geometry 265 5.3 Formal Axiomatics 275 5.4 Metamathematics 280 5.5 The Poincaré Model and the Consistency of Lobachevskian Plane 287 Geometry 5.6 Deductions from the Poincaré Model (Optional) 294 5.7 Plane Projective Geometry and a NonDesarguesian Geometry 300 5.8 Finite Geometries (Optional) 306 Bibliography 313 Appendix B: Euclid's First Principles and the Statements of the 317 Propositions of Book 1 Appendix C: Hilbert's Postulates for Plane Euclidean Geometry 323 Appendix D: The BirkhoffBeatley Postulates for Plane Euclidean 327 Geometry Suggestions for Solutions of Selected Problems 329 Index 367 Page ix PREFACE To prepare oneself to teach high school geometry, one should do two things: (1) learn considerably more elementary geometry than is covered in the high school course itself, (2) become acquainted with modern notions of geometric structure. The necessity of the first requirement follows from the wellknown pedagogical fact that one cannot give a strong, interesting, and inspiring course in a subject without knowing a good deal more about the subject than that which one teaches. The necessity of the second requirement follows from the fact that one cannot attain a true appreciation of presentday geometry without knowing something of the evolution of geometric structure from the days of the ancient Greeks up to modern times. The above observations led to the planning of this text, wherein Part I is devoted to modern elementary geometry and Part II to the evolution of geometric structure. It was felt that Part I should be a sequel to the basic geometry of the high school course and introduce the reader to some of the important modern extensions of elementary geometry—extensions that have largely entered into the mainstream of mathematics. On the other hand, Part II should treat notions of geometric structure via the nonEuclidean revolution that arose in the first half of the nineteenth century. It is hoped that such a twopart text can respectively furnish suitable introductions to (socalled) college geometry on the one hand and to nonEuclidean geometry on the other. It is through these two avenues that teachers should better be able to prepare themselves to rescue high school geometry from the doldrums of its present plight, and to rebreathe into it some of its inherent romance, beauty, and excitement. The text is adaptable to either a twosemester or a onesemester course. For a twosemester course, which is much the better option if time permits, essentially the whole text can be covered, with the exception of most or all of the optional sections. Greater discrimination will be required for a onesemester course. Here, in addition to the optional sections, the whole of Chapter 0 may have to be omitted, along with a number of the later sections of the other chapters, the omitted sections to be chosen by the Page x instructor. In any case, the optional sections make excellent assigned reading for the especially bright students in a class. Indeed, this is the chief reason for their inclusion. It has quite properly been said that "no subject loses more than does mathematics when dissociated from its history." The treatment in this book is strongly historical, for the study is concerned with fundamental ideas, and a genuine appreciation and understanding of ideas is not possible without an analysis of their origins. The history of a subject enlivens it, heightens its appeal, and brings out valuable cultural links. It is difficult to overstate the importance of problems in mathematics in general and geometry in particular. Problems are an integral part of geometry, for one learns geometry chiefly by doing geometry. This text contains an ample supply of problems, many of which introduce the student to interesting matter not appearing in the text proper. The student will miss much if he or she does not dip into the problem material. Since geometry problems often present difficulties to a beginner, suggestions for solutions of most of the problems are given at the end of the work. Finally, at the end of each chapter there is a short bibliography of related works. Page 1 0— The Fountainhead 0.1 The Earliest Geometry 0.2 The Empirical Nature of PreHellenic Geometry 0.3 The Greek Contribution of Material Axiomatics 0.4 Euclid's Elements 0.5 The Greek Bequest 0.6 The Transmission of Greek Geometry to the Occident 0.7 The Case for Empirical, or Experimental, Geometry (Optional) This first chapter is introductory in nature. It describes the source from which essentially all geometrical investigations of the modern era have arisen. This makes possible, in the succeeding chapters, an analysis of the origins of the fundamental concepts there introduced—an analysis without which neither a true understanding nor a genuine appreciation of these concepts is possible. The historical matter in this chapter is admittedly brief and sketchy. A reader interested in a considerably more detailed historical treatment can consult, for example, Howard Eves, An Introduction to the History of Mathematics (6th edition),* a work that has served as the chief basis for the attenuated sketch appearing here. 0.1— The Earliest Geometry The first geometrical considerations of man are unquestionably very ancient, and would seem to have their origin in simple observations stemming from human ability to recognize physical form and to compare shapes and sizes. *Saunders College Publishing, Philadelphia, 1990.
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