Euclidean and non-Euclidean geometries: development and history
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
San Francisco
W. H. Freeman and company
1980
|
| Edition: | second edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xv, 400 Seiten Illustrationen, Diagramme |
| ISBN: | 0716711036 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804117366594863104 |
|---|---|
| adam_text | Contents
Preface xi
Introduction 1
1 Euclid s Geometry
The origins of geometry 5
The axiomatic method 8
Undefined terms 9
Euclid s first four postulates 12
The parallel postulate 16
Attempts to prove the parallel postulate 18
Exercises 21
2 Logic
Informal logic 31
Theorems and proofs 32
RAA proofs 34
Negation 36
Quantifiers 38
Implication 40
Law of excluded middle 41
Incidence geometry 41
Models 43
Isomorphism of models 47
The danger in diagrams 48
Exercises 50
3 Hilbert s Axioms
Flaws in Euclid 57
Axioms of betweenness 59
Axioms of congruence 69
Axioms of continuity 78
Axiom of parallelism 82
Exercises 83
4 Neutral Geometry
Geometry without the parallel axiom 93
Alternate interior angle theorem 94
Exterior angle theorem 96
Measure of angles and segments 98
Saccheri Legendre theorem 100
Equivalence of parallel postulates 103
Angle sum of a triangle 105
Exercises 109
5 History of the Parallel Postulate
Proclus 119
Wallis 123
Saccheri and Lambert 125
Wolfgang Bolyai 127
Exercises 129
6 The Discovery of Non Euclidean Geometry
Janos Bolyai 140
Gauss 142
Lobachevsky 145
Hyperbolic geometry 147
Angle sums (again) 150
Similar triangles 150
Parallels that admit a common perpendicular 152
Limiting parallel rays 155
Classification of parallels 158
Strange new universe? 159
Exercises 160
7 Independence of the Parallel Postulate
Consistency of hyperbolic geometry 177
The Beltrami Klein model 180
The Poincare models 186
Perpendicularity in the Beltrami Klein model 192
Inversion in circles 194
The projective nature of the Beltrami Klein model 208
Exercises 219
8 Philosophical Implications
What is the geometry of physical space? 237
What is mathematics about? 239
The controversy about the foundations of
mathematics 241
The mess 245
Review Exercise 247
Some topics for essays 248
9 Geometric Transformations
Kleins Erlanger Programme 253
Groups 254
Applications to geometric problems 257
Motions and similarities 262
Reflections 264
Rotations 267
Translations 270
Half turns 273
Ideal points in the hyperbolic plane 274
Parallel displacements 276
Glides 277
Classification of motions 279
Automorphisms of the Cartesian model 282
Motions in the Poincare model 287
Congruence described by motions 295
Symmetry 298
Exercises 304
10 Further Results in Hyperbolic Geometry
Area and defect 319
The angle of parallelism 323
Cycles 324
The pseudosphere 326
Hyperbolic trigonometry 329
Circumference and area of a circle 337
Saccheri and Lambert quadrilaterals 340
Coordinates in the hyperbolic plane 346
The circumscribed cycle of a triangle 351
Exercises 356
Appendix A Elliptic and Other
Riemannian Geometries
Elliptic geometry 365
Riemannian geometry 370
Appendix B Geometry Without
Continuity 378
Suggested Further Reading 384
Bibliography 386
List of Axioms 390
List of Symbols 393
Name Index 395
Subject Index 397
|
| any_adam_object | 1 |
| author | Greenberg, Marvin J. 1935-2017 |
| author_GND | (DE-588)138411263 |
| author_facet | Greenberg, Marvin J. 1935-2017 |
| author_role | aut |
| author_sort | Greenberg, Marvin J. 1935-2017 |
| author_variant | m j g mj mjg |
| building | Verbundindex |
| bvnumber | BV003077579 |
| classification_rvk | SG 590 |
| ctrlnum | (OCoLC)630948245 (DE-599)BVBBV003077579 |
| discipline | Mathematik |
| edition | second edition |
| era | Geschichte gnd |
| era_facet | Geschichte |
| format | Book |
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| indexdate | 2024-07-09T15:53:16Z |
| institution | BVB |
| isbn | 0716711036 |
| language | English |
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| physical | xv, 400 Seiten Illustrationen, Diagramme |
| publishDate | 1980 |
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| publisher | W. H. Freeman and company |
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| spelling | Greenberg, Marvin J. 1935-2017 (DE-588)138411263 aut Euclidean and non-Euclidean geometries development and history second edition San Francisco W. H. Freeman and company 1980 xv, 400 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Geschichte gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Nichteuklidische Geometrie (DE-588)4042073-5 gnd rswk-swf Geschichte (DE-588)4020517-4 gnd rswk-swf Euklidische Geometrie (DE-588)4137555-5 gnd rswk-swf Geometrie (DE-588)4020236-7 s Geschichte (DE-588)4020517-4 s DE-604 Nichteuklidische Geometrie (DE-588)4042073-5 s Euklidische Geometrie (DE-588)4137555-5 s Geschichte z 1\p DE-604 2\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001929468&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 |
| spellingShingle | Greenberg, Marvin J. 1935-2017 Euclidean and non-Euclidean geometries development and history Geometrie (DE-588)4020236-7 gnd Nichteuklidische Geometrie (DE-588)4042073-5 gnd Geschichte (DE-588)4020517-4 gnd Euklidische Geometrie (DE-588)4137555-5 gnd |
| subject_GND | (DE-588)4020236-7 (DE-588)4042073-5 (DE-588)4020517-4 (DE-588)4137555-5 |
| 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 |
| title_fullStr | Euclidean and non-Euclidean geometries development and history |
| title_full_unstemmed | Euclidean and non-Euclidean geometries development and history |
| title_short | Euclidean and non-Euclidean geometries |
| title_sort | euclidean and non euclidean geometries development and history |
| title_sub | development and history |
| topic | Geometrie (DE-588)4020236-7 gnd Nichteuklidische Geometrie (DE-588)4042073-5 gnd Geschichte (DE-588)4020517-4 gnd Euklidische Geometrie (DE-588)4137555-5 gnd |
| topic_facet | Geometrie Nichteuklidische Geometrie Geschichte Euklidische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001929468&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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