Fundamental Concepts of Geometry (Dover Books on Mathematics)

Demonstrates in a clear and lucid manner the relationships between several types of geometry. This highly regarded work is a superior teaching text, especially valuable in teacher preparation, as well as providing an excellent overview of the foundations and historical evolution of geometrical concepts. Preface. Index. Bibliography. Exercises (no solutions). Includes 98 illustrations. Reprint of the Addison-Wesley Publishing Company, Reading, MA, 1955 edition. Table of Contents CHAPTER 1. FOUNDATIONS OF GEOMETRY 1-1 Logical systems 1-2 Logical notations 1-3 Inductive and deductive reasoning 1-4 Postulates 1-5 Independent postulates 1-6 Categorical sets of postulates 1-7 A geometry of number triples 1-8 Geometric invariants CHAPTER 2. SYNTHETIC PROJECTIVE GEOMETRY 2-1 Postulates of incidence and existence 2-2 Properties of a projective plane 2-3 Figures 2-4 Duality 2-5 Perspective figures 2-6 Projective transformations 2-7 Postulate of Projectivity 2-8 Quadrangles 2-9 Complete and simple n-points 2-10 Theorem of Desargues 2-11 Theorem of Pappus 2-12 Conics 2-13 Theorem of Pascal 2-14 Survey CHAPTER 3. COORDINATE SYSTEMS 3-1 Quadrangular sets 3-2 Properties of quadrangular sets 3-3 Harmonic sets 3-4 Postulates of Separation 3-5 Nets of rationality 3-6 Real projective geometry 3-7 Nonhomogeneous coordinates 3-8 Homogeneous coordinates 3-9 Survey CHAPTER 4. ANALYTIC PROJECTIVE GEOMETRY 4-1 Representations in space 4-2 Representations on a plane 4-3 Representations on a line 4-4 Matrices 4-5 Cross ratio 4-6 Analytic and synthetic geometries 4-7 Groups 4-8 Classification of projective transformations 4-9 Polarities and conics 4-10 Conics 4-11 Involutions on a line 4-12 Survey CHAPTER 5. AFFINE GEOMETRY 5-1 Ideal points 5-2 Parallels 5-3 Mid-point 5-4 Classification of conics 5-5 Affine transformations 5-6 Homothetic transformations 5-7 Translations 5-8 Dilations 5-9 Line reflections 5-10 Equiaffine and equiareal transformations 5-11 Survey CHAPTER 6. EUCLIDEAN PLANE GEOMETRY 6-1 Perpendicluar lines 6-2 Similarity transformations 6-3 Orthogonal line reflections 6-4 Euclidean transformations 6-5 Distances 6-6 Directed angles 6-7 Angles 6-8 Common figures 6-9 Survey CHAPTER 7. THE EVOLUTION OF GEOMETRY 7-1 Early measurements 7-2 Early Greek influence 7-3 Euclid 7-4 Early euclidean geometry 7-5 The awakening in Europe 7-6 Constructions 7-7 Descriptive geometry 7-8 Seventeenth ce 7-9 Eighteenth century 7-10 Euclid's fifth postulate 7-11 Nineteenth and twentieth centuries 7-12 Survey CHAPTER 8. NONEUCLIDEAN GEOMETRY 8-1 The absolute polarity 8-2 Points and lines 8-3 Hyperbolic geometry 8-4 Elliptic and spherical geometries 8-5 Comparisons CHAPTER 9. TOPOLOGY 9-1 Topology 9-2 Homeomorphic figures 9-3 Jordan Curve Theorem 9-4 Surfaces 9-5 Euler's Formula 9-6 Tranversable networks 9-7 Four-color problem 9-8 Fixed-point theorems 9-9 Moebius strip 9-10 Survey BIBLIOGRAPHY INDEX Cover......Page 1 S Title......Page 2 FUNDAMENTAL CONCEPTS OF GEOMETRY......Page 4 [QA445.M45 1983] 516......Page 5 PREFACE......Page 6 CONTENTS......Page 8 1-1 Logical systems......Page 12 ExERCI SES......Page 14 1-2 Logical notation......Page 15 EXERCISES......Page 17 1-3 Inductive and deductive reasoning......Page 18 1-4 Postulates......Page 20 EXERCISES......Page 22 1-5 Independent postulates......Page 23 EXERCISES......Page 24 1-6 Categorical sets of postulates......Page 25 1-7 A geometry of number triples......Page 29 EXERCISES......Page 31 1-8 Geometric invariants......Page 32 EXERCISES......Page 34 REVIEW EXERCISES......Page 35 CHAPTER 2 SYNTHETIC PROJECTIVE GEOMETRY......Page 36 2-1 Postulates of incidence and existence......Page 37 2-2 Properties of a projective plane......Page 40 2-3 Figures......Page 43 EXERCISES......Page 47 2-4 Duality......Page 48 ExERcIsEs......Page 50 2-5 Perspective figures......Page 51 2-6 Projective transformations......Page 54 EXERCISES......Page 60 2-7 Postulate of Projectivity......Page 61 2-8 Quadrangles......Page 63 EXERCISES......Page 66 2-9 Complete and simple n-points......Page 67 2-10 Theorem of Desargues......Page 69 2-11 Theorem of Pappus......Page 72 2-12 Conics......Page 74 EXERCISES......Page 76 2-13 Theorem of Pascal......Page 77 2-14 Survey......Page 78 REVIEW EXERCISES......Page 79 3-1 Quadrangular sets......Page 80 EXERCISES......Page 83 3-2 Properties of Quadrangular Sets......Page 84 3-3 Harmonic sets......Page 87 EXERCISES......Page 90 3-4 Postulates of Separation......Page 91 3-5 Nets of rationality......Page 94 3-6 Real projective geometry......Page 97 3-7 Nonhomogeneous coordinates......Page 103 EXERCISES......Page 105 3-8 Homogeneous coordinates......Page 106 EXERCISES......Page 110 3-9 Survey......Page 111 REVIEW EXERCISES......Page 112 4-1 Representations in space......Page 114 EXERCISES......Page 117 4-2 Representations on a plane......Page 118 EXERCISES......Page 120 4-3 Representations on a line.......Page 121 EXERCISES......Page 124 4-4 Matrices......Page 125 EXERCISES......Page 131 4-5 Cross ratio......Page 133 EXERCISES......Page 135 4-6 Analytic and synthetic geometries......Page 136 4-7 Groups......Page 139 EXERCISES......Page 141 4-8 Classification of projective transformations......Page 143 4-9 Polarities and conics......Page 146 EXERCISES......Page 150 4-10 Conics......Page 151 4-11 Involutions on a line......Page 155 EXERCISES......Page 157 4-12 Survey......Page 158 REVIEW EXERCISES......Page 159 5-1 Ideal points......Page 161 5-2 Parallels......Page 163 5-3 Mid-point......Page 166 EXERCISES......Page 169 5-4 Classification of conics......Page 170 5-5 Affine transformations......Page 173 EXERCISES......Page 176 5-6 Homothetic transformations......Page 177 5-7 Translations......Page 180 EXERCISES......Page 181 5-8 Dilation......Page 183 EXERCISES......Page 184 5-9 Line reflections......Page 185 5-10 Equiaffine and equiareal transformations......Page 188 EXERCISES......Page 191 5-11 Survey......Page 192 REVIEW EXERCISES......Page 194 6-1 Perpendicular lines.......Page 196 EXERCISES......Page 199 6-2 Similarity transformations......Page 200 6-3 Orthogonal line reflections......Page 202 6-4 Eucidean transformations......Page 205 EXERcISES......Page 209 6-5 Distances......Page 210 EXERCISES......Page 213 6-6 Directed angles......Page 214 EXERCISES......Page 219 6-7 Angles......Page 220 EXERCISES......Page 223 6-8 Common figures......Page 224 EXERCISES......Page 225 6-9 Survey......Page 226 REVIEW EXERCISES......Page 228 7-1 Early measurements......Page 230 7-2 Early Greek influence......Page 232 7-3 Euclid......Page 239 7-4 Early eudidean geometry......Page 246 7-5 The awakening in Europe......Page 251 7-6 Constructions......Page 253 7-7 Descriptive geometry......Page 255 7-8 Seventeenth century......Page 258 7-9 Eighteenth century......Page 261 7-10 Euclid's fifth postulate......Page 262 7-11 Nineteenth and twentieth centuries......Page 270 7-12 Survey......Page 276 8-1 The absolute polarity......Page 279 8-2 Points and lines......Page 283 8-3 Hyperbolic geometry......Page 285 EXERCISES......Page 291 8-4 Elliptic and spherical geometries......Page 292 8-5. Comparisons......Page 295 9-1 Topology......Page 299 EXERCISES......Page 302 9-2 Homeomorphic figures......Page 303 9-3 Jordan Curve Theorem......Page 305 9-4 Surfaces......Page 308 EXERCISES......Page 311 9-5 Euler's Formula......Page 312 EXERCISES......Page 313 9-6 Traversable networks......Page 314 EXERCISES......Page 315 9-7 Four-color problem......Page 316 9-8 Fixed-point theorems......Page 317 9-9 Moebius strip......Page 318 9-10 Survey......Page 320 BIBLIOGRAPHY......Page 322 INDEX......Page 324