Foundations of geometry: Euclidean and Bolyai-Lobachevskian geometry, projective geometry
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam
North-Holland
1960
|
| Ausgabe: | Rev. Engl. transl. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Aus d. Poln. übers. |
| Beschreibung: | XIV, 444 S. Ill. |
Internformat
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| 245 | 1 | 0 | |a Foundations of geometry |b Euclidean and Bolyai-Lobachevskian geometry, projective geometry |c by Karol Borsuk and Wanda Szmielew. Transl. from Polish by Erwin Marquit |
| 250 | |a Rev. Engl. transl. | ||
| 264 | 1 | |a Amsterdam |b North-Holland |c 1960 | |
| 300 | |a XIV, 444 S. |b Ill. | ||
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Datensatz im Suchindex
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|---|---|
| adam_text | CONTENTS
Contents v
Preface to the Polish Edition xi
Preface to the English Edition xm
Introduction
1. Geometry before Euclid 1
2. Elements of Euclid 1
3. Elementary Geometry after Euclid. Euclid s Critics and Commen¬
tators 2
4. Bolyai Lobachevskian Geometry 4
5. Consistency of Geometry 4
6. Riemann Spaces 6
7. Axiomatic Theory 6
8. Sets and Relations 7
9. Topological Space 11
10. Analytic Geometry 15
Part One
EUCLIDEAN AND BOLYAI LOBACHEVSKIAN GEOMETRY
Introduction to Part One
Primitive Notions and Axioms 19
Chapter I. Axioms of Incidence and Order
1. Axioms of Incidence 21
2. Three Non Collinear Points and Four Non Coplanar Points .... 22
3. Lines and Planes 22
4. Fundamental Existence Theorems 23
5. Intersections of Lines and Planes 25
6. Linear Axioms of Order 26
7. Segments. Open Segments 27
8. Open Segments on a Line 28
9. Division of an Open Segment 30
10. Common Part of Two Open Segments ¦ 31
11. Topology on the Line 32
12. Half Lines 33
13. Half Lines on a Line 36
14. Half Lines on a Given Half Line 37
15. Orientations of a Line. Axes 37
16. Order of Points on an Axis 39
17. Betweenness Relation for a Line and Two Points 41
}I CONTENTS
18. Plane Axiom of Order 42
19. Half Planes 43
20. Pencils and Half Pencils of Half Lines 47
21. Betweenness Relation for Half Lines 50
22. Ordering of a Half Pencil 55
23. Angles 55
24. Characterization of Lines and Planes in Terms of the Betweenness
Relation 57
25. Triangles. Open Triangles 58
26. The Open Triangle and the Line 61
27. Topology on the Plane 64
28. Polygonal Lines 68
29. Quadrangles 70
30. The Intersection Point of the Diagonals of a Quadrangle 71
31. Polygons and Their Triangulation 76
{hapter II. Axioms of Congruence
1. Axioms of Congruence. Congruence of Segments. Congruence of Figures 80
2. Relations Between Segments on Two Lines 82
3. Relations Between Segments on Two Planes 83
4. Congruence of Angles 84
5. Adjacent Angles. Vertical Angles 86
6. Relations Between Angles of Two Half Pencils 88
7. Relations Between Sides and Angles of Two Triangles 90
8. Relations Between Sides and Angles of a Triangle 91
9. The Relations Less Than and Greater Than for Segments 92
10. The Relations Less Than and Greater Than for Angles 94
11. Midpoint of a Segment 95
12. Bisector of an Angle 97
13. External Angles of a Triangle 97
14. Two Non Intersecting Lines on a Plane 98
15. Relations Between Sides and Angles of a Triangle (Conclusion) . . . 100
16. Relations Between Sides and Angles of Two Triangles (Continued) . 101
17. Free Segments. The Relations Less Than and Greater Than for Free
Segments. Addition of Free Segments 103
18. Subtraction of Free Segments. Multiplication of Free Segments by
Dyadic Numbers 105
19. Triangle Inequality 107
20. Free Angles. Calculus of Free Angles 109
21. Addition of Free Angles in a Pencil 112
22. The Sum of Two Angles of a Triangle 113
23. Right, Acute, and Obtuse Angles 114
24. Right, Acute, and Obtuse Free Angles 116
25. Right, Acute, and Obtuse Triangles 117
26. Perpendicular Lines 119
27. Perpendicular Projection upon a Line 120
28. Perpendicular Bisector of a Segment 121
29. The Saccheri Quadrangle 121
30. Rectangels 123
31. Line Perpendicular to a Plane 124
32. Perpendicular Planes 129
33. Perpendicular Projection upon a Plane 130
34. Congruence of Two Lines and Two Planes. Perfect Homogeneity of
the Line and of the Plane 131
35. Parallel Half Lines 138
CONTENTS VII
36. Parallel Axes 149
37. Parallel Lines 150
Chapter III. Axiom of Continuity
1. Axiom of Continuity 151
2. The Archimedean Postulate 154
3. The Saccheri Quadrangle (Conclusion) 157
4. The Saccheri Legendre Theorem 158
5. Parallel Half Lines (Continued) 160
6. Parallel Axes (Continued) 161
7. Angle of Parallelism 161
8. Parallel Lines (Continued) 165
9. Measure of Segments 167
10. Measure of Angles 172
11. The Saccheri Legendre Theorem Formulated in Terms of Measure . . 174
12. Distance Between Two Points. Space S as a Metric Space 176
13. Distance of a Point from a Figure 176
14. A Characterization of Bstweenness and Equidistance Relations in
Terms of Distance 177
15. Similitudes 177
16. Topology in Space Induced by a Metric 179
17. Distance as a Continuous Function of Two Points 181
18. Coordinates on a Line. Metric Type of Lines 184
19. Absolute Coordinates on a Plane. Topological Type of Planes. . . . 185
20. Absolute Coordinates in Space. Topological Type of Space 189
21. Rectangular Coordinates 192
Chapter IV. Models of Absolute Geometry
1. Problems of Consistency, Independence, and Categoricity of an Axiom
System of Geometry. Interpretation. Model 194
2. The Cartesian Space Cn 197
3. The Cartesian Model. Consistency of Absolute Geometry 211
4. Independence of the Axiom of Continuity 218
5. The Cartesian Circle 225
6. Projective Space Pn 232
7. The Klein Beltrami Model 245
8. Formula for Distance in Klein Space Ka 259
9. Non Categoricity of Absolute Geometry. Euclidean Geometry. Bolyai
Lobachevskian Geometry 262
Chapter V. Euclidean Geometry
1. The Axiom of Euclid 264
2. The Sum of the Angles of a Triangle and of a Quadrangle 264
3. Parallel Lines (Conclusion) 264
4. Parallel Half Lines (Conclusion) 265
5. Two Angles with Respectively Parallel Sides 267
6. Parallel Projection upon a Line 268
7. Parallelograms 268
8. The Theorem of Thales 269
9. Similar Triangles 272
10. Pythagorean Theorem 273
11. Metric Type of Planes 273
12. Metric Type of Space 274
13. Categoricity of Euclidean Geometry 276
VIII CONTENTS
Chapter VI. Bolyai Lobachevskian Geometry
1. The Axiom of Bolyai Lobachevski 278
2. The Sum of the Angles of a Triangle and of a Quadrangle 278
3. Relations Between Sides and Angles of Two Triangles (Conclusion) . 279
4. Similitudes 280
5. Defect of a Triangle 280
6. Defect of a Polygon 283
7. Defect of a Plane Figure 284
8. Parallel Lines and Hyperparallel Lines 284
9. Pairs of Parallel Lines 286
10. Pairs of Hyperparallel Lines 290
11. Perpendicular Projection of a Line Upon a Line 293
12. The Line of Enclosure 298
13. Natural Basic Segment. Natural Measure of Segments 300
14. Three Parallel Lines 301
15. Perpendicular Bisectors of the Three Sides of a Triangle 302
16. The Lobachevskian Function II 304
17. The Infinite Right Triangle 305
18. The Horocycle 308
19. Tangents and Secants of the Horocycle 316
20. Arc of the Horocycle 318
21. Length of Arc of the Horocycle 319
22. Translations on a Plane. Length of Arc on a Translated Horocycle . 323
23. Length of Arc of the Horocycle (Conclusion) 327
24. Sectors of the Horocycle 329
25. Formula of the Lobachevskian Function 331
26. The Functions sin II and cos II 334
27. Right Triangles 335
28. Quadrangle with Three Right Angles 338
29. Rectangular Coordinates on a Plane 339
30. Beltrami Coordinates on a Plane. Formula for Distance 341
31. Categoricity of Bolyai Lobachevskian Geometry 344
Part Two
PROTECTIVE GEOMETRY
Introduction to Part Two
Primitive Notions and Axioms 349
Chapter VII. Axioms of Incidence and Order
1. Axioms of Incidence 350
2. Fundamental Existence Theorems 351
3. Intersections of Lines and Planes 353
4. Central Projection upon a Line. Perspective and Projective Transfor¬
mations 354
5. Central Projection upon a Plane 355
6. Triangles. Perspective Center and Perspective Axis of Two Triangles.
The Theorem of Desargues 356
7. Axioms of Order 365
8. Segments. Open Segments 367
9. Properties of Open Segments 368
10. The Triple of Open Sides of a Triangle 370
11. Model for the Euclidean Axioms of Incidence and Order in Projective
Geometry. Proper Points, Lines, and Planes 370
CONTENTS IX
12. Ordinary Triangles 373
13. Topology on the Line 375
14. Topology on the Plane 376
15. Quadrangles 377
16. Harmonic Quadruples 379
17. Permutations of Harmonic Quadruples 380
18. The Fourth Harmonic Point 382
19. Perspective Transformations of Harmonic Quadruples 386
20. Continuity of the Central Projectivity 387
21. Midpoint of a Segment 388
22. Natural Net 388
23. Integral Net 391
24. Dyadic Net 392
Chapter VIII. Axiom of Continuity
1. Axiom of Continuity 395
2. Dyadic Net (Conclusion) 396
3. Real Net 398
4. Cartesian Coordinates on a Line 401
5. Cartesian Coordinates on a Plane 401
6. Equation of the Set of Proper Points of a Proper Line on a Plane . . 405
7. Cartesian Coordinates in Space 406
8. Equation of the Set of Proper Points of a Proper Line in Space . . 408
9. Equation of the Set of Proper Points of a Proper Plane in Space . . 409
10. Projective Coordinates in Space 411
11. Equations of the Plane and of the Line in Space 412
12. The Relation of Division and the Cross Ratio 414
Chapter IX. Models of Projective Geometry
1. Problems of Consistency and Categoricity 417
2. Model (P). Consistency of Projective Geometry 418
3. Categoricity of Space Projective Geometry 420
4. The Ellipse. Some Theorems Concerning the Circle and the Ellipse . 421
5. The Limit of a Sequence of Circles 425
6. Problem of the Categoricity of Plane Projective Geometry. The Hilbert
Model 426
Index of Geometrical Symbols 437
Index 440
|
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| author | Borsuk, Karol 1905-1982 Szmielew, Wanda 1918-1976 |
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| bvnumber | BV008934246 |
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| discipline | Mathematik Philosophie |
| edition | Rev. Engl. transl. |
| format | Book |
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| indexdate | 2024-07-09T17:27:00Z |
| institution | BVB |
| language | English |
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| spelling | Borsuk, Karol 1905-1982 Verfasser (DE-588)1027621473 aut Podstawy geometrii Foundations of geometry Euclidean and Bolyai-Lobachevskian geometry, projective geometry by Karol Borsuk and Wanda Szmielew. Transl. from Polish by Erwin Marquit Rev. Engl. transl. Amsterdam North-Holland 1960 XIV, 444 S. Ill. txt rdacontent n rdamedia nc rdacarrier Aus d. Poln. übers. Grundlage (DE-588)4158388-7 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Geometrie (DE-588)4020236-7 s Grundlage (DE-588)4158388-7 s DE-604 Szmielew, Wanda 1918-1976 Verfasser (DE-588)1104150816 aut Marquit, Erwin Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005892184&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Borsuk, Karol 1905-1982 Szmielew, Wanda 1918-1976 Foundations of geometry Euclidean and Bolyai-Lobachevskian geometry, projective geometry Grundlage (DE-588)4158388-7 gnd Geometrie (DE-588)4020236-7 gnd |
| subject_GND | (DE-588)4158388-7 (DE-588)4020236-7 |
| title | Foundations of geometry Euclidean and Bolyai-Lobachevskian geometry, projective geometry |
| title_alt | Podstawy geometrii |
| title_auth | Foundations of geometry Euclidean and Bolyai-Lobachevskian geometry, projective geometry |
| title_exact_search | Foundations of geometry Euclidean and Bolyai-Lobachevskian geometry, projective geometry |
| title_full | Foundations of geometry Euclidean and Bolyai-Lobachevskian geometry, projective geometry by Karol Borsuk and Wanda Szmielew. Transl. from Polish by Erwin Marquit |
| title_fullStr | Foundations of geometry Euclidean and Bolyai-Lobachevskian geometry, projective geometry by Karol Borsuk and Wanda Szmielew. Transl. from Polish by Erwin Marquit |
| title_full_unstemmed | Foundations of geometry Euclidean and Bolyai-Lobachevskian geometry, projective geometry by Karol Borsuk and Wanda Szmielew. Transl. from Polish by Erwin Marquit |
| title_short | Foundations of geometry |
| title_sort | foundations of geometry euclidean and bolyai lobachevskian geometry projective geometry |
| title_sub | Euclidean and Bolyai-Lobachevskian geometry, projective geometry |
| topic | Grundlage (DE-588)4158388-7 gnd Geometrie (DE-588)4020236-7 gnd |
| topic_facet | Grundlage Geometrie |
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