The Foundations of Geometry and the Non-Euclidean Plane

524 Pages·1975·15.18 MB·English

by  George E. Martin (auth.)

Undergraduate Texts in Mathematics
Editors
S. Axler
F.W. Gehring
K.A. Ribet
Springer
New York
Berlin
Heidelberg
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Budapest
Hong Kong
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(continued after index)
George E. Martin
The Foundations of Geometry
and the Non-Euclidean Plane
Springer
George E. Martin
Department of Mathematics and Statistics
State University of New York at Albany
1400 Washington Avenue
Albany, New York 12222
U.S.A.
Editorial Board
s. Axler F.W. Gehring K.A. Ribet
Department of Department of Department of
Mathematics Mathematics Mathematics
Michigan State University University of Michigan University of California
East Lansing, MI 48824 Ann Arbor, MI 48109 at Berkeley
U.S.A. U.S.A. Berkeley, CA 94720
U.S.A.
Mathematics Subject Classification (1991): 51-01,51-03.
This book was originally published by Intext Educational Publishers.
Library of Congress Cataloging in Publication Data
Martin, George Edward, 1932-
The foundations of geometry and the non-Euclidean
plane.
(Undergraduate texts in mathematics)
Reprint. Originally published: New York: Intext
Educational Publishers, 1975.
Includes index.
1. Geometry-Foundations. 2. Geometry, Non-Euclidean.
I. Title. II. Series. III. Series: Intext series in
mathematics.
QA681.M34 1982 516' .1 82-728
© 1975 by Springer-Verlag New York, Inc.
Softcover reprint of the hardcover 1st edition 1975
All rights reserved. No part of this book may be translated or reproduced in any form
without the written permission from Springer-Verlag, 175 Fifth Avenue, New York
10010, U.S.A.
9 8 7 6 5 4 (Corrected fourth printing, 1998)
ISBN-I3: 978-1-4612-5727-1 e-ISBN-I3: 978-1-4612-5725-7
001: 10.1007/978-1-4612-5725-7
To Margaret
Contents
Preface xiii
Foreword to the Student xv
INTRODUCTION
1 EQUIVALENCE RELATIONS 2
1.1 Logic 2
1.2 Sets 4
1.3 Relations 5
1.4 Exercises 8
Graffiti 9
2 MAPPINGS 10
2.1 One-to-One and Onto 10
2.2 Composition of Mappings 15
2.3 Exercises 17
Graffiti 19
3 THE REAL NUMBERS 20
3.1 Binary Operations 20
3.2 Properties of the Reals 26
3.3 Exercises 31
Graffiti 33
viii CONTENTS
4 AXIOM SYSTEMS 34
4.1 Axiom Systems 34
4.2 Incidence Planes 36
4.3 Exercises 45
Graffiti 47
PART ONE ABSOLUTE GEOMETRY
5 MODELS 50
5.1 Models of the Euclidean Plane 50
5.2 Models of Incidence Planes 55
5.3 Exercises 61
Graffiti 64
6 INCIDENCE AXIOM AND RULER POSTULATE 65
6.1 Our Objectives 65
6.2 Axiom 1: The Incidence Axiom 66
6.3 Axiom 2: The Ruler Postulate 68
6.4 Exercises 70
Graffiti 72
7 BETWEENNESS 73
7.1 Ordering the Points on a Line 73
7.2 Taxicab Geometry 77
7.3 Exercises 81
Graffiti 82
8 SEGMENTS, RAYS, AND CONVEX SETS 84
8.1 Segments and Rays 84
8.2 Convex Sets 89
8.3 Exercises 92
Graffiti 93
9 ANGLES AND TRIANGLES 95
9.1 Angles and Triangles 95
9.2 More Models 100
9.3 Exercises 109
Graffiti 110
CONTENTS ix
10 THE GOLDEN AGE OF GREEK MATHEMATICS 111
(Optional)
10.1 Alexandria 111
10.2 Exercises 119
11 EUCLID'S ELEMENTS (Optional) 121
11.1 The Elements 121
11.2 Exercises 129
Graffiti 130
12 PASCH'S POSTULATE AND PLANE
SEPARATION POSTULATE 131
12.1 Axiom 3: PSP 131
12.2 Pasch, Peano, Pieri, and Hilbert 137
12.3 Exercises 140
Graffiti 142
13 CROSSBAR AND QUADRILATERALS 144
13.1 More Incidence Theorems 144
13.2 Quadrilaterals 149
13.3 Exercises 152
Graffiti 153
14 MEASURING ANGLES AND THE PROTRACTOR
POSTULATE 155
14.1 Axiom 4: The Protractor Postulate 155
14.2 Peculiar Protractors 166
14.3 Exercises 169
15 ALTERNATIVE AXIOM SYSTEMS (Optional) 172
15.1 Hilbert's Axioms 172
15.2 Pieri's Postulates 175
15.3 Exercises 180
16 MIRRORS 182
16.1 Rulers and Protractors 182
16.2 MIRROR and SAS 184
16.3 Exercises 189
Graffiti 191
X CONTENTS
17 CONGRUENCE AND THE PENULTIMATE
POSTULATE 192
17.1 Congruence for Triangles 192
17.2 Axiom 5: SAS 195
17.3 Congruence Theorems 198
17.4 Exercises 201
Graffiti 202
18 PERPENDICULARS AND INEQUALITIES 204
18.1 A Theorem on Parallels 204
18.2 Inequali ties 207
18.3 Right Triangles 211
18.4 Exercises 213
Graffiti 215
19 REFLECTIONS 216
19.1 Introducing Isometries 216
19.2 Reflection in a Line 219
19.3 Exercises 223
Graffiti 225
20 CIRCLES 226
20.1 Introducing Circles 226
20.2 The Two-Circle Theorem 230
20.3 Exercises 236
Graffiti 238
21 ABSOLUTE GEOMETRY AND SACCHERI
QUADRILATERALS 239
21.1 Euclid's Absolute Geometry 239
21.2 Giordano's Theorem 248
21.3 Exercises 252
Graffiti 253
22 SACCHERI'S THREE HYPOTHESES 255
22.1 Omar Khayyam's Theorem 255
22.2 Saccheri's Theorem 260
22.3 Exercises 266
Graffiti 267
CONTENTS xi
23 EUCLID'S PARALLEL POSTULATE 269
23.1 Equivalent Statements 269
23.2 Independence 281
23.3 Exercises 286
Graffiti 289
24 BIANGLES 292
24.1 Closed Biangles 292
24.2 Critical Angles and Absolute Lengths 295
24.3 The Invention of Non-Euclidean Geometry 302
24.4 Exercises 314
Graffiti 316
25 EXCURSIONS 317
25.1 Prospectus 317
25.2 Euclidean Geometry 320
25.3 Higher Dimensions 323
25.4 Exercises 328
Graffiti 330
PART TWO NON-EUCLIDEAN GEOMETRY
26 PARALLELS AND THE ULTIMATE AXIOM 334
26.1 Axiom 6: HPP 334
26.2 Parallel Lines 338
26.3 Exercises 344
Graffiti 346
27 BRUSHES AND CYCLES 347
27.1 Brushes 347
27.2 Cycles 351
27.3 Exercises 356
Graffiti 358
28 ROTATIONS, TRANSLATIONS, AND
HOROLATIONS 360
28.1 Products of Two Reflections 360
28.2 Reflections in Lines of a Brush 365
28.3 Exercises 368
Graffiti 370