Fundamental Concepts of Geometry

333 Pages·1983·8.62 MB·English

by  Bruce E. Meserve

FUNDAMENTAL
CONCEPTS
OF GEOMETRY

FUNDAME NTAL
CONCEPTS
OF GEOMETRY
BRUCE E. MESERVE
DOVER PUBLICATIONS, INC., NEW YORK
ri° 1955, 1983 by Bruce E. Meserve.
All rights reserved under Pan American and International Copyright
Conventions.
Published in Canada by General Publishing Company, Ltd., 30 Lesmill
Road, Don Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Company, Ltd.,
10 Orange Street, London WC2H 7EG.
This Dover edition, first published in 1983, is an unabridged and slightly
corrected republication of the second printing (1959) of the work origi-
nally published by Addison-Wesley Publishing Company, Inc., Reading,
Mass., in 1955.
Manufactured in the United States of America
Dover Publications, Inc., 180 Varick Street, New York, N.Y. 10014
Library of Congress Cataloging in Publication Data
Meserve, Bruce Elwyn, 1917-
Fundamental concepts of geometry.
Reprint. Originally published: Reading, Mass. : Addison-Wesley Pub.
Co., 1959. (Addison-Wesley mathematics series) With slight corrections.
Bibliography: p.
Includes index.
1. Geometry. I. Title. II. Series: Addison-Wesley mathematics series.
[QA445.M45 1983] 516 82-18309
ISBN 0-486-63415-9
PREFACE
This book and its companion volume * on algebra and analysis
are based upon a course entitled " Fundamental Concepts of Mathe-
matics " as it has evolved at the University of Illinois during the past
forty years. These two books provide a broad mathematical per-
spective for readers with a maturity equivalent to at least one year
and preferably two years of college mathematics. Neither book is
a prerequisite for the other. When both texts are to be used, the
author prefers to cover the algebra before the geometry, but many
of his students take the geometry first. Both texts have been used
in mimeographed form for several years at the University of Illinois.
There is ample material in each text for a course having 45 class
hours. Both texts reflect the recognition of a basic need for a knowl-
edge of fundamental concepts of mathematics apart from what is
gained in the specialized courses in each of its many subdivisions.
Prospective teachers of secondary mathematics, students preparing
for specialized advanced courses in mathematics, and those desiring
a broad liberal education have a particular need for the mathematical
perspective gained from an elementary treatment of the fundamental
concepts of mathematics.
The primary purpose of this book is to help the reader
(i) to discover how euclidean plane geometry is related to,
and often a special case of, many other geometries,
(ii) to obtain a practical understanding of "proof,"
(iii) to obtain the concept of a geometry as a logical system
based upon postulates and undefined elements, and
(iv) to appreciate the historical evolution of our geometrical
concepts and the relation of euclidean geometry to the space in
which we live.
These goals are sought through an introduction to the foundations
of geometry (Chapter 1) and a consideration of the following hier-
archy of geometries.
* Meserve, B. E., Fundamental Concepts of Algebra. Addison-Wesley
Publishing Company, Inc., 1953.
V
vi PREFACE
topology (9)
projective (2, 3, 4)
affine (5) noneuclidean (8)
euclidean (6, 7)
The numbers in parentheses indicate the chapters in which the
geometries are discussed. The use of both synthetic and algebraic
methods enables the reader to develop also an appreciation for each
of these important methods.
The fundamental concepts of euclidean geometry are treated in
the specialization of projective geometry to obtain euclidean geometry
(Chapters 2 through 6) . Chapters 7 (the evolution of geometry) , 8
(noneuclidean geometry), and 9 (topology) are introductions to topics
that will increase the reader's understanding of euclidean geometry.
Unfortunately these topics can not be treated thoroughly in the
space available.
Chapters 1 through 6 form a convenient unit for short courses in
which the discussion of conics (Sections 2—1 1, 2—12, 2—13, 4—9, 4—10,
and 4—1 1) and angles (Sections 6—6 and 6—7) may be omitted if
essary. Chapters 7 and 9 may be read independently either before
or after Chapters 1 through 6. Chapter 8 is based upon Chapters
1 through 7.
The author is deeply indebted to Professors J. W. Young, E. B.
Lytle, Echo Pepper, and J. H. Chanler for their part in the evolution
of the course on which this book is based; to the constructive criti-
cisms of many students; to his wife, who typed the manuscript; and
to the publishers for their cooperation and very efficient service. To
each and all the author is sincerely grateful.
Ii E.M.
July, 1954
CONTENTS
CHAPTER 1. FOUNDATIONS OF GEOMETRY I I I I I S 5 1
1—1 Logical systems . . . . I 1
S
Logical notations 4
1—2
1—3 Inductive and deductive reasoning . 7
1—4 Postulates I I I I • • I 9
.......
1—5 Independent postulates . . .1412
Categorical sets of postulates
1—6
1—7 A geometry of number triples 11111112118
Geometric invariants
1—8
CHAPTER 2. SYNTHETIC PxtosEcnvE GEOMETRY I I I I I I I 25
2—1 Postulates of incidence and existence I I 26
2-2 PFroipegrtieus orf ea psro.je.c.ti.ve. .pl.a.ne. 29
2—3 32
2-4 Duality 37
Perspective figures
2—5 . . . .
2—6 Projective transformations .
2—7 Postulate of Projectivity . . I • I
2-8 Quadrangles
Complete and simple n-points
2—9
2-10 Theorem of Desargues
. I I
2—li Theorem of Pappus . . .
2—12 Conics 1111111
Theorem of Pascal
2—13 11111. 11I I I I I
2—14 Survey
CawrzR 3.
COORDINATE SYSTEMS .
Quadrangular sets
3—1 . . . I
•
Properties of quadrangular sets
3—2 111111
3—3 Harmonic sets I 76
3—4 Postulates of Separa1tio1n1. 1I 1 80
•
Nets of rationality 83
3—5
3—6 Real projective geometry . I 86
Nonhomogeneous coordinates 92
3—7 I •IIIIII
3—8 Homoge1neo1us1 co1or1din1ate1s 1I1I 95
111100
Survey
3—9
CHAnR 4.
ANALYTIC PROJECTIVE G1EO1M1ET1RY1 I11I 1I 1I 1I 1I1I 103
4—1 Representations in space 1111111101703
4-2 Representations on a plane
I I I
vu
Viii CONTENTS
Representations on a line
4—3 . . . . • . 110
4—4 Matrices 114
4—5 Cross ratio 122
4—6 Analytic and synthetic geometries . 125
4—7 Groups 128
4—8 Classification of projective transformations • . . 132
4—9 Polarities and conics 135
4—10 Conics
4-11 Involutions on a line 144
.
4—12 Survey 147
CHAPTER 5. AFFINE GEOMETRY . , . . .
5—i Idealpoints 150
5-2 Parallels 152
5-2 Mid-point 155
5-4 Classification of conics 159
5—5 Affine transformations • . . . 162
5—6 Homothetic transformations . . . 166
5-7 Translations 169
5—8 Dilations 172
5—9 Line reflections 174
5—10 Equiaffine and equiareal transformations 177
5-41 Survey
CHAPTER 6. EUCLIDEAN PLANE GEOMETRY 185
6-1 Perpendicular lines 185
6—2 Similarity transformations . 189
6—3 Orthogonal line reflections . 191
6-4 Euclidean transformations 194
6—5 Distances 199
6—6 Directed angles 203
Angles 209
6—7
6-8 Common figures 213
Survey 215
6—9
CHAPTER 7. THE EVOLUTION OF GEOMETRY . 219
Early measurements 219
7—1
7—2 Early Greek influence 221
Euclid 228
7—3
7—4 Early euclidean geometry. . . 235
The awakening in Europe 240
7—5 . .
Constructions 242
7—6
.244
7—7 Descriptive geometry . . . .
CONTENTS ix
7—8 Seventeenth century 247
7—9 Eighteenth century 250
7—10 Euclid's fifth postulate 251
7—11 Nineteenth and twentieth centuries 259
7—12 Survey . . . . . . . 265
CaAFrER 8. NONEUCLIDEAN GEOMETRY 268
.268
The absolute polarity
8—1 . .
Points and lines 272
8—2
Hyperbolic geometry 274
8—3 I
.281
Elliptic and spherical geometries
8—4
8—5 Comparisons . . . I I I I I I I I I I I I 284
Cawrrnt 9. TOPOLOGY I I I I I I 288
9—1 Topology . . . . . • . . 288
Homeomorphic figures 292
9—2
Jordan Curve Theorem 294
9—3
9—4 Surfaces 297
9-5 Euler's Formula. 301
9—6 Traversable networks 303
Four-color problem. 305
9—7
Fixed-point theorems 306
9—8
9—9 Moebius strip 307
9—10 Survey 309
BIBLIOGRAPHY . . . . . I I I I I I I I I . . 311
INDEX . . . . . . . . . . . . . . . . . . . . . . 313