Euclidean and non-euclidean geometries: development and history

Marvin Jay Greenberg0716724464, 9780716724469

This classic text provides overview of both classic and hyperbolic geometries, placing the work of key mathematicians/ philosophers in historical context. Coverage includes geometric transformations, models of the hyperbolic planes, and pseudospheres.

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Table of contents :
Front cover……Page 1
Title page……Page 3
Date-line……Page 4
Dedication……Page 5
CONTENTS……Page 7
PREFACE……Page 11
INTRODUCTION……Page 19
The origins of geometry……Page 24
The axiomatic method……Page 28
Undefined terms……Page 29
Euclid’s first four postulates……Page 32
The parallel postulate……Page 36
Attempts to prove the parallel postulate……Page 39
The danger in diagrams……Page 41
The power of diagrams……Page 43
Review exercise……Page 44
Exercises……Page 45
Major exercises……Page 49
Projects……Page 53
Informal logic……Page 56
Theorems and proofs……Page 58
RAA proofs……Page 60
Negation……Page 62
Quantifiers……Page 63
Implication……Page 66
Law of excluded middle and proof by cases……Page 67
Incidence geometry……Page 68
Models……Page 70
Isomorphism of models……Page 74
Projective and affine planes……Page 76
Review exercise……Page 80
Exercises……Page 81
Major exercises……Page 83
Projects……Page 86
Flaws in Euclid……Page 88
Axioms of betweenness……Page 90
Axioms of congruence……Page 100
Axioms of continuity……Page 111
Axiom of parallelism……Page 120
Review exercise……Page 121
Exercises……Page 122
Major exercises……Page 129
Projects……Page 132
Geometry without the parallel axiom……Page 133
Alternate interior angle theorem……Page 134
Exterior angle theorem……Page 136
Measure of angles and segments……Page 140
Saccheri-Legendre theorem……Page 142
Equivalence of parallel postulates……Page 146
Angle sum of a triangle……Page 148
Review exercise……Page 152
Exercises……Page 154
Major exercises……Page 161
Projects……Page 164
5 HISTORY OF THE PARALLEL POSTULATE……Page 166
Proclus……Page 167
Wallis……Page 169
Saccheri……Page 172
Clairaut……Page 174
Legendre……Page 175
Lambert and Taurinus……Page 177
Farkas Bolyai……Page 179
Review exercise……Page 181
Exercises……Page 182
Major exercises……Page 192
Projects……Page 194
Janos Bolyai……Page 195
Gauss……Page 198
Lobachevsky……Page 201
Subsequent developments……Page 203
Hyperbolic geometry……Page 205
Similar triangles……Page 207
Parallels that admit a common perpendicular……Page 209
Limiting parallel rays……Page 213
Classification of parallels……Page 216
Strange new universe?……Page 218
Review exercise……Page 219
Exercises……Page 221
Major exercises……Page 227
Projects……Page 239
Consistency of hyperbolic geometry……Page 241
The Beltrami-Klein model……Page 245
The Poincare models……Page 250
Perpendicularity in the Beltrami-Klein model……Page 256
A model of the hyperbolic plane from physics……Page 259
Inversion in circles……Page 261
The projective nature of the Beltrami-Klein model……Page 276
Review exercise……Page 288
K-Exercises……Page 289
P-Exercises……Page 297
H-Exercises……Page 304
What is the geometry of physical space?……Page 308
What is mathematics about?……Page 311
The controversy about the foundations of mathematics……Page 313
The mess……Page 317
Review exercise……Page 319
Some topics for essays……Page 320
Klein1s Erlanger Programme……Page 327
Groups……Page 329
Applications to geometric problems……Page 333
Motions and similarities……Page 339
Reflections……Page 342
Rotations……Page 345
Translations……Page 348
Half-turns……Page 351
Ideal points in the hyperbolic plane……Page 352
Parallel displacements……Page 355
Glides……Page 356
Classification of motions……Page 358
Automorphisms of the Cartesian model……Page 362
Motions in the Poincare model……Page 367
Congruence described by motions……Page 376
Symmetry……Page 381
Review exercise……Page 387
Exercises……Page 390
Area and defect……Page 404
The angle of parallelism……Page 409
Cycles……Page 410
The pseudosphere……Page 412
Hyperbolic trigonometry……Page 416
Circumference and area of a circle……Page 425
Saccheri and Lambert quadrilaterals……Page 429
Coordinates in the hyperbolic plane……Page 435
The circumscribed cycle of a triangle……Page 441
Review exercise……Page 446
Exercises……Page 447
Elliptic geometry……Page 456
Riemannian geometry……Page 461
Appendix B Geometry Without Continuity……Page 472
Suggested Further Reading……Page 479
Bibliography……Page 481
List of Axioms……Page 487
List of Symbols……Page 490
Name Index……Page 492
Subject Index……Page 496
Back cover……Page 502