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Methods of Geometry PDF

507 Pages·2011·7.95 MB·English
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Methods of Geometry James T. Smith San Francisco State University A Wiley-lnterscience Publication JOHN WILEY & SONS, INC. New York · Chichester · Weinheim · Brisbane · Singapore · Toronto This page intentionally left blank Methods of Geometry This page intentionally left blank Methods of Geometry James T. Smith San Francisco State University A Wiley-lnterscience Publication JOHN WILEY & SONS, INC. New York · Chichester · Weinheim · Brisbane · Singapore · Toronto This book is printed on acid-free paper. ® Copyright © 2000 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750- 4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: [email protected] For ordering and customer service, call 1-800-CALL WILEY. Library of Congress Cataloging In Publication Data: Smith, James T., 1939- Methods of geometry / James T. Smith, p. cm. "A Wiley-lnterscience publication." Includes bibliographical references. ISBN 0-471-25183-6 (alk. paper) I. Geometry. I. Title. QA445.S57 2000 516—dc21 98-10306 10 9 8 7 6 5 4 3 21 This book is dedicated to the place where it was conceived, in the Wind River Mountains. The nearest official landmark is indicated by a cross on a topographical map. That refers to a little brass plate embedded in a rock about the size of your desk. The plate reads U S Geological Survey for information write the Director Washington, DC Cooperation with the State Elevation Above /+\ sea 10342 feet GWM8 1938 Benchmark The map, based on the 1937-1938 survey, locates the benchmark at 42°54.08' Ν, 109°21.40' W. It's in the Jim Bridger Wilder- ness, in Wyoming. That world is like geometry: tranquil, but grand; lonely, but populated by wonders. Entering it is an adventure. You'll experience fascination, challenge, and appre- hension. And when you become comfortable there, you'll feel great joy. This page intentionally left blank Contents Preface xi About the author xv 1 Introduction 1 1.1 Episodes 2 1.2 Advanced geometry 9 1.3 This book 13 1.4 Reading about geometry 16 1.5 Projects 17 2 Foundations 19 2.1 Geometry as applied mathematics 20 2.2 Need for rigor 23 2.3 Axiomatic method 26 2.4 Euclid's Elements 29 2.5 Coordinate geometry 33 2.6 Foundation problem 35 2.7 Parallel axiom 37 2.8 Firm foundations 39 2.9 Geometry as pure mathematics 42 2.10 Exercises and projects 50 3 Elementary Euclidean geometry 53 3.1 Incidence geometry 55 3.2 Ruler axiom and its consequences 57 3.3 Pasch's axiom and the separation theorems 60 3.4 Angles and the protractor axioms 64 3.5 Congruence 67 3.6 Perpendicularity 71 3.7 Parallel axiom and related theorems 76 3.8 Area and Pythagoras' theorem 81 3.9 Similarity 89 3.10 Polyhedral volume 92 3.11 Coordinate geometry 100 3.12 Circles and spheres 106 3.13 Arcs and trigonometric functions 110 3.14 π 119 vii

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