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