A Survey of Geometry, Revised Edition

Cover......Page 1
Title page......Page 3
Contents......Page 7
PREFACE TO THE REVISED EDITION......Page 11
PREFACE TO THE FIRST EDITION......Page 13
The Earliest Geometry......Page 23
The Empirical Nature of Pre-Hellenic Geometry......Page 25
The Greek Contribution of Material Axiomatics......Page 30
Euclid's "Elements"......Page 38
The Geometrical Contributions of Euclid and Archimedes......Page 43
Apollonius and Later Greek Geometers......Page 50
The Transmission of Greek Geometry to the Occident......Page 60
The Case for Empirical, or Experimental, Geometry......Page 67
2. MODERN ELEMENTARY GEOMETRY......Page 75
Sensed Magnitudes......Page 76
Infinite Elements......Page 81
The Theorems of Menelaus and Ceva......Page 85
Applications of the Theorems of Menelaus and Ceva......Page 89
Cross Ratio......Page 95
Applications of Cross Ratio......Page 98
Homographic Ranges and Pencils......Page 101
Harmonic Division......Page 104
Orthogonal Circles......Page 109
The Radical Axis of a Pair of Circles......Page 114
3. ELEMENTARY TRANSFORMATIONS......Page 121
Transformation Theory......Page 122
Fundamental Point Transformations of the Plane......Page 126
Applications of the Homothety Transformation......Page 129
Isometries......Page 135
Similarities......Page 139
Inversion......Page 142
Properties of inversion......Page 148
Applications of inversion......Page 152
Reciprocation......Page 161
Applications of Reciprocation......Page 165
Space Transformations......Page 169
4. EUCLIDEAN CONSTRUCTIONS......Page 176
The Euclidean Tools......Page 177
The Method of Loci......Page 180
The Method of Transformation......Page 183
The Double Points of Two Coaxial Homographic Ranges......Page 188
The Mohr-Mascheroni Construction Theorem......Page 191
The Poncelet-Steiner Construction Theorem......Page 196
Some Other Results......Page 201
The Regular Seventeen-Sided Polygon......Page 207
5. DISSECTION THEORY......Page 216
Preliminaries......Page 218
Dissection of Polygonsnto Triangles......Page 223
The Fundamental Theorem of Polygonal Dissection......Page 227
Lennes Polyhedra and Cauchy's Theorem......Page 233
Dehn's Theorem......Page 235
Congruency (T) and Suss' Theorem......Page 240
Congruency by Decomposition......Page 243
A Brief Budget of Dissection Curiosities......Page 248
6. PROJECTIVE GEOMETRY......Page 262
Perspectivities and Projectivities......Page 264
Further Applications......Page 269
Proper Conics......Page 273
Applications......Page 277
The Chasles-Steiner Definition of a Proper Conic......Page 281
A Proper Conic as an Envelope of Lines......Page 285
Reciprocation and the Principle of Duality......Page 288
The Focus-Directrix Property......Page 294
Orthogonal Projection......Page 297
7. NON-EUCLIDEAN GEOMETRY......Page 304
Historical Background......Page 305
Parallels and Hyperparallels......Page 313
Limit Triangles......Page 317
Saccheri Quadrilaterals and the Angle-Sum of a Triangle......Page 320
Area of a Triangle......Page 323
Ideal and Ultra-Ideal Points......Page 326
An Application of Ideal and Ultra-Ideal Points......Page 330
Mapping the Plane onto thenterior of a Circle......Page 333
Geometry and Physical Space......Page 336
8. THE FOUNDATIONS OF GEOMETRY......Page 341
Some Logical Shortcomings of Euclid's "Elements"......Page 342
Modern Postulational Foundations for Euclidean Geometry......Page 349
Formal Axiomatics......Page 359
Metamathematics......Page 365
The Poincaré Model and the Consistency of Lobachevskian Plane Geometry......Page 369
Deductions from the Poincaré Model......Page 375
A Postulational Foundation for Plane Projective Geometry......Page 381
Non-Desarguesian Geometry......Page 384
Finite Geometries......Page 387
APPENDIX 1. Euclid's First Principles and the Statements of the Propositions of Book......Page 397
APPENDIX 2. Hilbert's Postulates for Plane Euclidean Geometry......Page 402
SUGGESTIONS FOR SOLUTIONS of Some of the Problems......Page 405
INDEX......Page 447