Elementary Geometry from an Advanced Standpoint (3rd Edition)

514 Pages·1990·13.51 MB·English

by  Edwin Moise

THIRD(cid:9) EDITION
V
ELEMENTARY GEOMETRY
from an
Advanced Standpoint
Edwin E. Moise
Emeritus, Queens College of the City University of New York
•
7'
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Library of Congress Cataloging-in-Publication Data
Moise, Edwin E.
Elementary geometry from an advanced standpoint / by Edwin E.
Moise. —3rd ed.
P.(cid:9) cm.
ISBN 0-201-50867-2
1. Geometry.(cid:9) I. Title.
QA445.M58 1990(cid:9) 89-37241
516.2 — dc20(cid:9) CIP
Photos on pages 92, 138, 249, 416, 454
Courtesy of The Bettman Archive.
Copyright © 1990 by Addison-Wesley Publishing Company, Inc.
All rights reserved. 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, or otherwise, without the prior written permission of the publisher. Printed in
the United States of America.
3 4 5 6 7 8 9 10 DO 9594939291
Preface
The title of this book is the best brief description of its content and purposes
that the author was able to think of. These purposes are rather different from
those of most books on "higher geometry" or the foundations of geometry.
The difference that is involved here is somewhat like the difference between
the two types of advanced calculus courses now commonly taught. Some courses
in advanced calculus teach material which has not appeared in the preceding
courses at all. There are others which might more accurately be called courses in
elementary calculus from an advanced standpoint: their purpose is to clean up
behind the introductory courses, furnishing valid definitions and valid proofs
for concepts and theorems which were already known, at least in some sense
and in some form. One of the purposes of the present book is to reexamine
geometry in the same spirit.
If we grant that elementary geometry deserves to be thoroughly understood,
then it is plain that such a job needs to be done; and no such job is done in any
college course now widely taught. The usual senior-level courses in higher ge-
ometry proceed on the very doubtful assumption that the foundations are well
understood. And courses in the foundations (when they are taught at all) are
usually based on such delicate postulate sets, and move so slowly, that they
cover little of the substance of the theory. The upshot of this is that mathemat-
ics students commonly leave college with an understanding of elementary
geometry which is not much better than the understanding that they acquired
in high school.
The purpose of this book is to elucidate, as thoroughly as possible, both this
elementary material and its surrounding folklore. My own experience, in teach-
ing the course to good classes, indicates that it is not safe to presuppose an
exact knowledge of anything. Moreover, the style and the language of traditional
geometry courses are rather incongruous with the style and the language of
the rest of mathematics today. This means that ideas which are, essentially, well
understood may need to be reformulated before we proceed. (See, for example,
Chapter 6, on Congruences Between Triangles.) In some cases, the reasons for
reformulation are more compelling. For example, the theory of geometry in-
equalities is used in the chapter on hyperbolic geometry; and this would hardly
iv(cid:9) Preface
be reasonable if the student had not seen these theorems proved without the
use of the Euclidean parallel postulate.
For these reasons, the book begins at the beginning. Some of the chapters
are quite easy, for a strong class, and can simply be assigned as outside reading.
Others, such as Chapters 20 and 24, are more difficult. These differences are
due to the nature of the material: it is not always possible to get from one place
to another by walking along a path of constant slope.
This unevenness in level of difficulty makes the book rather flexible. A well-
prepared class may go rapidly through Chapters 1-7, and devote most of its
time to the sort of material presented in Chapters 8, 10, 14, 19, and 20. A
poorly prepared class may go carefully through Chapter 1-7, omit such chap-
ters as 14 and 20, and still not get to Chapter 25.
The book is virtually self-contained. The necessary fragments of the theory
of equations and the theory of numbers are presented in Chapters 28 and 29,
at the end. At many points, ideas from algebra and analysis are needed in the
discussion of the geometry. These ideas are explained in full, on the ground
that it is easier to skip explanations of things that are known than to find con-
venient and readable references. The only exception to this is in Chapter 22,
where it seemed safe to assume that epsilon-delta limits are understood.
In some chapters, especially Chapters 20 and 25, we give full expositions of
topics which are commonly dismissed rather briefly and almost casually. The
process by which the real numbers are introduced into an Archimedean geome-
try, for purposes of measurement, is highly important and far from trivial. The
same is true of the consistency proof for the hyperbolic postulates. Here (as
elsewhere) the purpose of this book is to explain, with all possible lucidity and
thoroughness, ideas which are widely alluded to but not so widely understood.
This third edition differs from the first and second in the following ways.
1. I have added biographical notes on various eminent mathematicians:
Euclid, Descartes, Lobachevsky, Hilbert, and Birkhoff, with the emphasis on
their contributions to geometry.
2. Section 4.6, The Seven Bridges of Konigsberg, is new.
3. Section 13.6, The Euclidian Program, continued: Equal-area without Area,
is new.
4. Chapter 28 of the first edition gave an example of an ordered field
which is not Archimedean. The example is more significant if the field is Eu-
clidian, that is, if every positive element has a square root. The new Chapter 32
makes the example Euclidian.
I believe that these changes are improvements; and am sure, at least, none
of them are deleterious.
The manuscript of this edition was reviewed, thoroughly and thoughtfully,
by Chris Coray, Utah State University; Richard Rogers, Weber State University; James
Slifker, Florida International University; Jerry L. Young, Boise State University.
I have had the benefit of their comments, and am accordingly grateful.
E. E. M.
September 1989
Contents
Chapter 1 The Algebra of the Real Numbers 1
1.1 Introduction(cid:9) 1
1.2 Addition and multiplication of real numbers(cid:9) 1
1.3 Fields(cid:9) 10
1.4 The ordering of the real numbers(cid:9) 12
1.5 Order relations and ordered fields(cid:9) 21
1.6 The positive integers and the induction principle(cid:9) 23
1.7 The integers and the rational numbers(cid:9) 28
1.8 The Archimedean postulate; Euclidean completeness(cid:9) 31
1.9 The language and notation of sets(cid:9) 35
1.10 n-fold sums and products; the generalized associative law(cid:9) 37
1.11 The closure of the integers under addition and multiplication(cid:9) 40
Chapter 2 Incidence Geometry in Planes and Space 43
Chapter 3 Distance and Congruence 48
3.1 The idea of a function(cid:9) 48
3.2 The set-theoretic interpretation of functions and relations(cid:9) 53
3.3 The distance function(cid:9) 56
3.4 Betweenness(cid:9) 60
3.5 Segments, rays, angles, and triangles(cid:9) 64
3.6 Congruence of segments(cid:9) 68
Chapter 4 Separation in Planes and Space 72
4.1 Convexity and separation(cid:9) 72
4.2 Incidence theorems based on the plane-separation postulate(cid:9) 76
4.3 Incidence theorems, continued(cid:9) 80
4.4 Convex quadrilaterals(cid:9) 83
4.5 Separation of space by planes(cid:9) 85
4.6 The seven bridges of Konigsberg(cid:9) 87
Leonard Euler (cid:9) 92
vi (cid:9) Contents
Chapter 5 Angular Measure 93
Chapter 6 Congruences Between Triangles 100
6.1 The idea of a congruence 100
6.2 The basic congruence theorems 104
6.3 Some remarks on terminology 110
6.4 The independence of the SAS postulate 112
6.5 Existence of perpendiculars 114
Chapter 7 (cid:9) Geometric Inequalities (cid:9) 116
Chapter 8 (cid:9) The Euclidean Program: Congruence Without Distance(cid:9) 125
8.1 The synthetic postulates 125
8.2 The laws of inequalities for segments 128
8.3 Right angles, synthetically considered 132
8.4 The synthetic form of the triangular inequality. Addition of
congruence classes 135
8.5 A summary of the differences between the metric and synthetic
approaches 137
Euclid 138
George David Birkhoff 138
Chapter 9(cid:9) Three Geometries(cid:9) 139
9.1 Introduction 139
9.2 The Poincare model for Lobachevskian geometry 140
9.3 The spherical model for Riemannian geometry 144
9.4 Some questions for later investigation 146
Chapter 10 (cid:9) Absolute Plane Geometry (cid:9) 148
10.1 Sufficient conditions for parallelism 148
10.2 The polygonal inequality 151
10.3 Saccheri quadrilaterals 152
10.4 The basic inequality for angle-sums in a triangle 155
10.5 A historical comedy 158
Chapter 11(cid:9) The Parallel Postulate and Parallel Projection(cid:9) 160
11.1 The uniqueness of parallels 160
11.2 Parallel projections 163
11.3 The comparison theorem 166
11.4 The basic similarity theorem 167
Contents vii
Chapter 12 (cid:9) Similarities Between Triangles(cid:9) 172
12.1 Proportionalities 172
12.2 Similarities between triangles 174
12.3 The Pythagorean theorem 178
Chapter 13(cid:9) Polygonal Regions and Their Areas(cid:9) 184
13.1 The area postulates 184
13.2 Area theorems for triangles and quadrilaterals 187
13.3 Applications of area theory: a simple proof of the basic
similarity theorem 193
13.4 Further applications of area theory: the Pythagorean theorem 195
13.5 A weaker form of the unit postulate A-5 198
13.6 The Euclidean program continued: Equal-area without area 200
Chapter 14 (cid:9) The Construction of an Area Function(cid:9) 203
14.1 The problem 203
14.2 Complexes and their formula areas 204
14.3 Verification of the area postulates for the function a 210
Chapter 15(cid:9) Perpendicular Lines and Planes in Space(cid:9) 212
15.1 The Basic theorems 212
15.2 Parallel lines and planes in space 218
15.3 The measure of a dihedral angle: perpendicular lines 221
Chapter 16 (cid:9) Circles and Spheres(cid:9) 224
16.1 Basic definitions 224
16.2 Secant and tangent lines. The line-circle theorem 225
16.3 Secant and tangent planes 229
16.4 Arcs of circles 231
16.5 The two-circle theorem 237
Chapter 17(cid:9) Cartesian Coordinate Systems (cid:9) 243
Rene Descartes 249
Chapter 18(cid:9) Rigid Motion(cid:9) 251
18.1 The most general concept of congruence 251
18.2 Isometries between triangles 254
18.3 General properties of isometries. Reflections 257
18.4 Dilations and similarities 261
viii (cid:9) Contents
Chapter 19 Constructions with Ruler and Compass 264
19.1 Unmarked rulers and collapsible compasses(cid:9) 264
19.2 How to do algebra with ruler and compass(cid:9) 268
19.3 Solving equations with ruler and compass(cid:9) 271
19.4 The problem of Apollonius(cid:9) 275
19.5 The impossible construction problems of antiquity(cid:9) 278
19.6 The surd field(cid:9) 279
19.7 The surd plane(cid:9) 282
19.8 Quadratic extension of fields. Conjugates in a quadratic
extension field(cid:9) 283
19.9 Surd fields of order n; surds of order n(cid:9) 288
19.10 Applications to cubic equations with rational coefficients(cid:9) 289
19.11 The trisection of the angle(cid:9) 291
19.12 The duplication of the cube(cid:9) 294
Chapter 20 From Eudoxus to Dedekind 295
20.1 Proportionalities without numbers(cid:9) 295
20.2 Eudoxus' synthetic definition of proportionality(cid:9) 297
20.3 The algebra of segment addition(cid:9) 301
20.4 How to define ratios: the supremum(cid:9) 303
20.5 Proof of the addition theorem(cid:9) 307
20.6 The metrization theorem(cid:9) 313
20.7 The Dedekind cut(cid:9) 318
Chapter 21 Length and Plane Area 320
21.1 The definition of arc length(cid:9) 320
21.2 The existence of 7F(cid:9) 322
21.3 Limits as the mesh approaches zero(cid:9) 325
21.4 The addition theorem for arc length(cid:9) 328
21.5 Approaching the area of a circular sector from below(cid:9) 330
21.6 Approaching the area of a circular sector from above(cid:9) 333
21.7 The area of a sector(cid:9) 335
Chapter 22 Jordan Measure in the Plane 336
22.1 The basic definition(cid:9) 336
22.2 The class of measurable sets(cid:9) 339
22.3 Areas under the graphs of continuous functions(cid:9) 348
Chapter 23 Solid Mensuration: The Elementary Theory 352
23.1 Basic assumptions for the theory of volume(cid:9) 352
23.2 Cross sections of cones and pyramids(cid:9) 354
23.3 Prisms and cylinders(cid:9) 358
23.4 Volumes of prisms and cylinders(cid:9) 360
Contents (cid:9) ix
23.5 Volumes of pyramids and cones(cid:9) 363
23.6 The volume of a sphere(cid:9) 366
Chapter 24 Hyperbolic Geometry 370
24.1 Absolute geometry, continued: the critical function(cid:9) 370
24.2 Absolute geometry: open triangles and critically parallel rays(cid:9) 376
24.3 Hyperbolic geometry: closed triangles and angle sums(cid:9) 382
24.4 Hyperbolic geometry: the defect of a triangle and the collapse
of similarity theory(cid:9) 386
24.5 Absolute geometry: triangulations and subdivisions(cid:9) 389
24.6 Euclidean geometry: Bolyai's theorem(cid:9) 394
24.7 Hyperbolic area-theory: the defect of a polygonal region(cid:9) 398
24.8 Bolyai's theorem for triangles in the hyperbolic case(cid:9) 401
24.9 Defects of small triangles(cid:9) 405
24.10 The continuity of the defect(cid:9) 407
24.11 Bolyai's theorem for polygonal regions in the hyperbolic case(cid:9) 410
24.12 The impossibility of Euclidean area-theory in hyperbolic
geometry(cid:9) 411
24.13 The uniqueness of hyperbolic area theory(cid:9) 413
24.14 Alternative forms of the parallel postulate(cid:9) 415
Nikolai Ivanovitch Lobachevsky(cid:9) 416
Chapter 25 The Consistency of the Hyperbolic Postulates 417
25.1 Introduction(cid:9) 417
25.2 Inversions of a punctured plane(cid:9) 417
25.3 Preservation of the cross ratio under inversions(cid:9) 422
25.4 Preservation of angular measure under inversions(cid:9) 423
25.5 Reflections across L-lines in the Poincare model(cid:9) 427
25.6 Uniqueness of the L-line through two points(cid:9) 429
25.7 The ruler postulate, betweenness and plane separation(cid:9) 431
25.8 Angular measure in the Poincare model(cid:9) 435
25.9 The SAS postulate(cid:9) 435
Chapter 26 The Consistency of Euclidean Geometry 438
26.1 Introduction(cid:9) 438
26.2 The ruler postulate(cid:9) 439
26.3 Incidence and parallelism(cid:9) 440
26.4 Translations and rotations(cid:9) 442
26.5 Plane separation(cid:9) 447
26.6 Angle congruences(cid:9) 449
David Hilbert(cid:9) 454