Basic concepts of geometry:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Blaisdell
1965
|
| Edition: | 1. ed. |
| Series: | A Blaisdell book in the pure and applied sciences
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 350 S. |
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MARC
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| 100 | 1 | |a Prenowitz, Walter |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Basic concepts of geometry |c Walter Prenowitz ; Meyer Jordan |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Blaisdell |c 1965 | |
| 300 | |a XIX, 350 S. | ||
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| 490 | 0 | |a A Blaisdell book in the pure and applied sciences | |
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Record in the Search Index
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| adam_text | Contents
PART I
INTRODUCTION 1
Chapter 1. LOGICAL DEFICIENCIES IN EUCLIDEAN
GEOMETRY 3
1. Proof That Every Triangle Is Isosceles 4
2. Where Does the Difficulty Lie 5
3. A Triangle with Two Right Angles 6
4. Proof of a Theorem 7
5. Another Flawed Proof 10
6. Summary 13
7. Implications for Teaching 14
exercises 14
references 19
Chapter 2. EUCLID S PARALLEL POSTULATE 20
1. The Structure of Euclidean Plane Geometry 21
2. A Substitute for Euclid s Parallel Postulate 25
3. The Equivalence of Euclid s and Playfair s Postulates 25
4. The Role of Euclid s Parallel Postulate 27
5. Proclus Proof of Euclid s Parallel Postulate 28
6. Wallis Solution of the Problem 29
xiii
Xiv CONTENTS
7. Saccheri s Attempt to Vindicate Euclid 31
exercises 32
references 36
Chapter 3. NEUTRAL GEOMETRY 37
1. Introduction 37
2. The Sum of the Angles of a Triangle 38
3. Do Rectangles Exist 40
4. Again, the Sum of the Angles of a Triangle 45
PROPOSITIONS OF PLANE NEUTRAL GEOMETERY 46
EXERCISES 48
Chapter 4. INTRODUCTION TO NON EUCLIDEAN
GEOMETRY 53
1. Lobachevskian Geometry 53
2. A Nonmetrical Theorem 54
3. An Objection 56
4. The Angle Sum of a Triangle in Lobachevskian Geometry 57
5. Do Similar Triangles Exist in Lobachevskian Geometry 62
6. Lobachevskian Theory of Area 63
7. Parallelism and Equidistance of Lines 66
8. Are There Other Neutral Geometries 68
9. Riemann s Non Euclidean Theory of Geometry 72
10. Lines as Closed Figures 75
11. Representation on a Euclidean Sphere 76
12. Critique 78
13. Representation of Mathematical Systems 79
14. Difficulties Encountered in a Formal Treatment of Riemann s
Theory 80
15. Polar Properties in Elliptic Plane Geometry 81
16. Further Remarks on Elliptic Geometry 84
17. Conclusion 84
exercises 86
references 89
Chapter 5. THE LOGICAL CONSISTENCY OF THE
NON EUCLIDEAN GEOMETRIES 90
1. Is Lobachevskian Geometry Consistent 90
2. The Representation 91
CONTENTS XV
3. The Conversion Rule 92
4. An Objection 93
5. The Scale of the Representation 94
6. The Property that a Lobachevskian Line is Metrically Infinite 95
7. Incompleteness of the Discussion 98
8. Verification of Lobachevskian Theorems 98
9. Consistency Proof for Lobachevskian Geometry 99
10. Further Remarks on Consistency 100
exercises 101
references 104
Chapter 6. THE EMPIRICAL VALIDITY OF THE
NON EUCLIDEAN GEOMETRIES 105
1. How is Geometry Applied Practically 106
2. Attempt to Verify a Postulate 107
3. Can the Parallel Postulate be Verified 108
4. The Angle Sum of a Triangle as a Crucial Property 108
5. Refinement of the Physical Processes 110
6. Analysis of the Angle Sum Property of Lobachevskian
Geometry 110
7. Lobachevskian Geometry in the Small as an Approximation
to Euclidean 111
8. Spherical Geometry in the Small as an Approximation to
Euclidean 113
9. Other Crucial Properties 115
10. Conclusion 116
PART II
INTRODUCTION 117
Chapter 7. THEORY OF INCIDENCE 119
1. The Concept of Incidence 119
2. Incidence in Terms of the Set Concept 120
3. Postulates for Incidence 121
4. Some Remarks on the Postulates 121
5. A Note on Language 123
6. Elementary Theorems on Incidence 124
7. The Role of Intuition 127
XVi CONTENTS
8. Parallel Planes and Parallel Lines 128
9. Existence Theorems 130
exercises 131
Chapter 8. INCIDENCE GEOMETRIES—MODELS OF
THE THEORY OF INCIDENCE 136
1. The Abstract Nature of the Theory of Incidence 136
2. Interpretation of the Theory of Incidence 137
3. Incidence Geometries 139
exercises 146
4. Discussion of the Algebraic Model M19 148
exercises 149
5. The Concept of Isomorphism 150
EXERCISES 152
6. Isomorphic and Automorphic Correspondences 153
7. Some Additional Models 154
exercises 155
8. The Use of Models to Prove Consistency 156
9. The Use of Models to Prove Independence 157
exercises 158
10. Existence of Points, Lines, Planes 159
11. Two Types of Incidence Geometry 160
exercises 161
Chapter 9. THEORY OF AFFINE GEOMETRIES 167
1. Introduction 167
2. The On Language for Incidence 168
3. Parallelism of Lines 168
4. Transversality of Lines 171
5. Transversality of Lines and Planes 172
6. Transversality of Planes 175
7. Conditions for Parallelism of Planes 176
EXERCISES 178
Chapter 10. THEORY OF ORDER ON THE LINE 184
1. The Concept of Order 184
2. Postulates for Betweenness 185
3. Elementary Properties of Betweenness 186
CONTENTS XVii
4. Segments 187
5. Rays or Half Lines 188
6. Concepts and Terminology of Set Theory 190
7. Decomposition of a Line Determined by Two of its Points 193
8. Determination of Rays 193
9. Opposite Rays 197
10. The Concept of Separation 198
11. Separation of a Line by One of its Points 200
12. Separation of a Segment by One of its Points 203
13. Convex Sets 205
14. Uniqueness of Opposite Ray 206
15. Extension of the Concept of Order 208
16. Models of the Theory 210
exercises 210
Chapter 11. PLANAR AND SPATIAL ORDER
PROPERTIES 216
1. Introduction 216
2. Ordered Incidence Geometries 217
3. Half Planes 218
4. Determination of Half Planes 220
5. Opposite Half Planes 223
6. The Separation Concept Again 225
7. Separation of a Plane by One of Its Lines 225
8. Convexity of Half Planes 227
9. Uniqueness of Opposite Half Plane 228
10. Spatial Theory of Order 230
11. Half Spaces 231
12. Determination of Half Spaces 233
13. Opposite Half Spaces 234
14. Spatial Separation by a Plane 232
15. Convexity of Half Spaces 234
16. Uniqueness of Opposite Half Space 234
exercises 235
Chapter 12. ANGLES AND ORDER OF RAYS 240
1. Introduction 240
2. The Sides of a Line 241
xviii contents
3. Betweenness of Rays 244
4. Relation Between Order of Points and Order of Rays 247
5. Order Properties Involving Opposite Rays 249
6. The Notion of Angle 254
exercises 257
Chapter 13. SEPARATION PROPERTIES OF ANGLES
AND TRIANGLES 263
1. The Interior of an Angle 263
2. Formula for a Half Plane 265
3. Formula for a Plane 266
4. The Exterior of an Angle 267
5. Separation and the Notion of Path 268
6. Separation of a Plane by an Angle 269
7. Triangles 272
8. The Interior of a Triangle 272
9. The Exterior of a Triangle 274
10. Separation of a Plane by a Triangle 276
exercises 280
Chapter 14. THEORY OF CONGRUENCE—INTRODUCTION
TO EUCLIDEAN GEOMETRY 287
1. Introduction 288
2. Postulates for Distance 289
3. Distance and Order of Points 289
4. Distance Functions 291
5. Postulates for Angle Measure 292
6. Angle Measure and Order of Rays 294
exercises 295
7. Congruence of Segments 295
8. Congruence of Angles 296
9. Supplementary Angles 298
10. Right Angles and Perpendicularity 300
11. Congruence of Triangles 301
12. Some Elementary Consequences of the SAS Postulate 303
13. Equidistance and Perpendicular Bisectors of Segments 304
14. The Exterior Angle Theorem 307
15. The Perpendicular to a Line from an External Point 309
CONTENTS Xix
16. Alternate Interior Angles and Parallel Lines 310
17. Euclidean Geometry 312
exercises 314
Chapter 15. CONGRUENCE WITHOUT NUMBERS—AN
INTRODUCTION 316
1. Postulates for Congruence 316
2. Order of Segments 318
exercises 319
3. Addition of Segments 319
exercises 321
4. Order of Angles 321
exercises 322
5. Addition of Angles 323
exercises 324
6. Congruence of Triangles 324
7. Supplementary Angles, Right Angles, Vertical Angles 325
8. Equidistance and Perpendicular Bisectors of Segments 327
9. The Exterior Angle Theorem and its Consequences 333
10. Construction of Distance Functions 334
exercises 337
list of postulates 339
Index 343
|
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| edition | 1. ed. |
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| series2 | A Blaisdell book in the pure and applied sciences |
| spelling | Prenowitz, Walter Verfasser aut Basic concepts of geometry Walter Prenowitz ; Meyer Jordan 1. ed. New York [u.a.] Blaisdell 1965 XIX, 350 S. txt rdacontent n rdamedia nc rdacarrier A Blaisdell book in the pure and applied sciences Géométrie - Fondements Geometry Foundations Jordan, Meyer Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007823922&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Prenowitz, Walter Jordan, Meyer Basic concepts of geometry Géométrie - Fondements Geometry Foundations |
| title | Basic concepts of geometry |
| title_auth | Basic concepts of geometry |
| title_exact_search | Basic concepts of geometry |
| title_full | Basic concepts of geometry Walter Prenowitz ; Meyer Jordan |
| title_fullStr | Basic concepts of geometry Walter Prenowitz ; Meyer Jordan |
| title_full_unstemmed | Basic concepts of geometry Walter Prenowitz ; Meyer Jordan |
| title_short | Basic concepts of geometry |
| title_sort | basic concepts of geometry |
| topic | Géométrie - Fondements Geometry Foundations |
| topic_facet | Géométrie - Fondements Geometry Foundations |
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