Euclidean and non-Euclidean geometries: development and history
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Freeman
1993
|
| Ausgabe: | 3. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 483 S. Ill., graph.Darst. |
| ISBN: | 0716724464 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804123887605121024 |
|---|---|
| adam_text | CONTENTS
PREFACE xi
INTRODUCTION I
1 EUCLID S GEOMETRY 6
The origins of geometry 6
The axiomatic method 10
Undefined terms 11
Euclid s first four postulates 14
The parallel postulate 18
Attempts to prove the parallel postulate 21
The danger in diagrams 23
The power of diagrams 25
Review exercise 26 Exercises 27
Major exercises 31 Projects 35
2 LOGIC AND INCIDENCE GEOMETRY 38
Informal logic 38
Theorems and proofs 40
RAA proofs 42
Negation 44
Quantifiers 45
Implication 48
Law of excluded middle and proof by cases 49
Incidence geometry 50
Models 52
Isomorphism of models 56
Projective and affine planes 58
Review exercise 62 Exercises 63
Major exercises 65 Projects 68
3 HILBERT S AXIOMS 70
Flaws in Euclid 70
Axioms of betweenness 72
Axioms of congruence 82
Axioms of continuity 93
Axiom of parallelism 102
Review exercise 103 Exercises 104
Major exercises 111 Projects 114
4 NEUTRAL GEOMETRY 115
Geometry without the parallel axiom 115
Alternate interior angle theorem 116
Exterior angle theorem 118
Measure of angles and segments 122
Saccheri Legendre theorem 124
Equivalence of parallel postulates 128
Angle sum of a triangle 130
Review exercise 134 Exercises 136
Major exercises 143 Projects 146
5 HISTORY OF THE PARALLEL POSTULATE 148
Proclus 149
Wallis 151
Saccheri 154
Clairaut 156
Legendre 157
Lambert and Taurinus 159
Farkas Bolyai 161
Review exercise 163 Exercises 164
Major exercises 174 Projects 176
6 THE DISCOVERY OF NON EUCLIDEAN
GEOMETRY 177
Janos Bolyai 177
Gauss 180
Lobachevsky 183
Subsequent developments 185
Hyperbolic geometry 187
Angle sums (again) 189
Similar triangles 189
Parallels that admit a common perpendicular 191
Limiting parallel rays 195
Classification of parallels 198
Strange new universe? 200
Review exercise 201 Exercises 203
Major exercises 209 Projects 221
7 INDEPENDENCE OF THE PARALLEL
POSTULATE 223
Consistency of hyperbolic geometry 223
The Beltrami Klein model 227
The Poincare models 232
Perpendicularity in the Beltrami Klein model 238
A model of the hyperbolic plane from physics 241
Inversion in circles 243
The projective nature of the Beltrami Klein mode! 258
Review exercise 270 K Exercises 271
P Exercises 279 H Exercises 286
8 PHILOSOPHICAL IMPLICATIONS 290
What is the geometry of physical space? 290
What is mathematics about? 293
The controversy about the foundations of mathematics 295
The mess 299
Review exercise 301 Some topics for essays 302
9 GEOMETRIC TRANSFORMATIONS 309
Klein s Erlanger Programme 309
Groups 311
Applications to geometric problems 315
Motions and similarities 321
Reflections 324
Rotations 327
Translations 330
Half turns 333
Ideal points in the hyperbolic plane 334
Parallel displacements 336
Glides 338
Classification of motions 340
Automorphisms of the Cartesian model 344
Motions in the Poincare model 349
Congruence described by motions 358
Symmetry 363
Review exercise 369 Exercises 372
10 FURTHER RESULTS IN HYPERROLIC
GEOMETRY 386
Area and defect 386
The angle of parallelism 391
Cycles 392
The pseudosphere 394
Hyperbolic trigonometry 398
Circumference and area of a circle 407
Saccheri and Lambert quadrilaterals 411
Coordinates in the hyperbolic plane 417
The circumscribed cycle of a triangle 423
Review exercise 428 Exercises 429
Appendix A Elliptic and Other Riemannian
Geometries 428
Elliptic geometry 438
Riemannian geometry 443
Appendix B Geometry Without Continuity 454
Suggested Further Reading 461
Bibliography 463
List of Axioms 469
List of Symbols 472
Name Index 474
Subject Index 478
|
| any_adam_object | 1 |
| author | Greenberg, Marvin J. |
| author_facet | Greenberg, Marvin J. |
| author_role | aut |
| author_sort | Greenberg, Marvin J. |
| author_variant | m j g mj mjg |
| building | Verbundindex |
| bvnumber | BV009547344 |
| callnumber-first | Q - Science |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SG 590 |
| ctrlnum | (OCoLC)27727445 (DE-599)BVBBV009547344 |
| dewey-full | 516 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516 |
| dewey-search | 516 |
| dewey-sort | 3516 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. ed. |
| era | Geschichte gnd |
| era_facet | Geschichte |
| format | Book |
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| spelling | Greenberg, Marvin J. Verfasser aut Euclidean and non-Euclidean geometries development and history Marvin Jay Greenberg 3. ed. New York Freeman 1993 XVI, 483 S. Ill., graph.Darst. txt rdacontent n rdamedia nc rdacarrier Geschichte gnd rswk-swf Geometria (história) larpcal Geometria não euclidiana (história) larpcal Meetkunde gtt Geschichte Geometry Geometry History Geometry, Non-Euclidean Geometry, Non-Euclidean History Geometrie (DE-588)4020236-7 gnd rswk-swf Nichteuklidische Geometrie (DE-588)4042073-5 gnd rswk-swf Euklidische Geometrie (DE-588)4137555-5 gnd rswk-swf Geschichte (DE-588)4020517-4 gnd rswk-swf Geometrie (DE-588)4020236-7 s Geschichte (DE-588)4020517-4 s DE-604 Geschichte z DE-188 Euklidische Geometrie (DE-588)4137555-5 s Nichteuklidische Geometrie (DE-588)4042073-5 s 1\p DE-604 2\p DE-604 3\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006306706&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Greenberg, Marvin J. Euclidean and non-Euclidean geometries development and history Geometria (história) larpcal Geometria não euclidiana (história) larpcal Meetkunde gtt Geschichte Geometry Geometry History Geometry, Non-Euclidean Geometry, Non-Euclidean History Geometrie (DE-588)4020236-7 gnd Nichteuklidische Geometrie (DE-588)4042073-5 gnd Euklidische Geometrie (DE-588)4137555-5 gnd Geschichte (DE-588)4020517-4 gnd |
| subject_GND | (DE-588)4020236-7 (DE-588)4042073-5 (DE-588)4137555-5 (DE-588)4020517-4 |
| title | Euclidean and non-Euclidean geometries development and history |
| title_auth | Euclidean and non-Euclidean geometries development and history |
| title_exact_search | Euclidean and non-Euclidean geometries development and history |
| title_full | Euclidean and non-Euclidean geometries development and history Marvin Jay Greenberg |
| title_fullStr | Euclidean and non-Euclidean geometries development and history Marvin Jay Greenberg |
| title_full_unstemmed | Euclidean and non-Euclidean geometries development and history Marvin Jay Greenberg |
| title_short | Euclidean and non-Euclidean geometries |
| title_sort | euclidean and non euclidean geometries development and history |
| title_sub | development and history |
| topic | Geometria (história) larpcal Geometria não euclidiana (história) larpcal Meetkunde gtt Geschichte Geometry Geometry History Geometry, Non-Euclidean Geometry, Non-Euclidean History Geometrie (DE-588)4020236-7 gnd Nichteuklidische Geometrie (DE-588)4042073-5 gnd Euklidische Geometrie (DE-588)4137555-5 gnd Geschichte (DE-588)4020517-4 gnd |
| topic_facet | Geometria (história) Geometria não euclidiana (história) Meetkunde Geschichte Geometry Geometry History Geometry, Non-Euclidean Geometry, Non-Euclidean History Geometrie Nichteuklidische Geometrie Euklidische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006306706&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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