# The Foundations of Geometry and the Non-Euclidean Plane

Springer Science & Business Media, 19‏/12‏/1997 - 512 من الصفحات
This book is a text for junior, senior, or first-year graduate courses traditionally titled Foundations of Geometry and/or Non Euclidean Geometry. The first 29 chapters are for a semester or year course on the foundations of geometry. The remaining chap ters may then be used for either a regular course or independent study courses. Another possibility, which is also especially suited for in-service teachers of high school geometry, is to survey the the fundamentals of absolute geometry (Chapters 1 -20) very quickly and begin earnest study with the theory of parallels and isometries (Chapters 21 -30). The text is self-contained, except that the elementary calculus is assumed for some parts of the material on advanced hyperbolic geometry (Chapters 31 -34). There are over 650 exercises, 30 of which are 10-part true-or-false questions. A rigorous ruler-and-protractor axiomatic development of the Euclidean and hyperbolic planes, including the classification of the isometries of these planes, is balanced by the discussion about this development. Models, such as Taxicab Geometry, are used exten sively to illustrate theory. Historical aspects and alternatives to the selected axioms are prominent. The classical axiom systems of Euclid and Hilbert are discussed, as are axiom systems for three and four-dimensional absolute geometry and Pieri's system based on rigid motions. The text is divided into three parts. The Introduction (Chapters 1 -4) is to be read as quickly as possible and then used for ref erence if necessary.

### ما يقوله الناس -كتابة مراجعة

لم نعثر على أي مراجعات في الأماكن المعتادة.

### المحتوى

 Equivalence Relations 11 LOGIC 2 12 SETS 4 13 RELATIONS 5 14 EXERCISES 8 GRAFFITI 9 Mappings 21 ONETOONE AND ONTO 10 22 COMPOSITION OF MAPPINGS 15 23 EXERCISES 17
 Reflections 191 INTRODUCING ISOMETRIES 216 192 REFLECTION IN A LINE 219 193 EXERCISES 223 GRAFFITI 225 Circles 201 INTRODUCING CIRCLES 226 202 THE TWOCIRCLE THEOREM 230 203 EXERCISES 236 GRAFFITI 238

 GRAFFITI 19 The Real Numbers 31 BINARY OPERATIONS 20 32 PROPERTIES OF THE REALS 26 33 EXERCISES 31 GRAFFITI 33 Axiom Systems 41 AXIOM SYSTEMS 34 42 INCIDENCE PLANES 36 43 EXERCISES 45 GRAFFITI 47 Models 51 MODELS OF THE EUCLIDEAN PLANE 50 52 MODELS OF INCIDENCE PLANES 55 53 EXERClSES 61 GRAFFITI 64 Incidence Axiom and Ruler Postulate 61 OUR OBJECTIVES 65 THE INCIDENCE AXIOM 66 THE RULER POSTULATE 68 64 EXERCISES 70 GRAFFITI 72 Betweenness 71 ORDERING THE POINTS ON A LINE 73 72 TAXICAB GEOMETRY 77 73 EXERCISES 81 GRAFFITI 82 Segments Rays and Convex Sets 81 SEGMENTS AND RAYS 84 82 CONVEX SETS 89 83 EXERCISES 92 GRAFFITI 93 Angles and Triangles 91 ANGLES AND TRIANGLES 95 92 MORE MODELS 100 93 EXERCISES 109 GRAFFITI 110 The Golden Age of Greek Mathematics 101 ALEXANDRIA 111 102 EXERCISES 119 Euclids Elements 111 THE ELEMENTS 121 112 EXERCISES 129 GRAFFITI 130 Paschs Postulate and Plane Separation Postulate 121 AXIOM 3 PSP 131 122 PASCH PEANO PIERI AND HILBERT 137 123 EXERCISES 140 GRAFFITI 142 Crossbar and Quadrilaterals 131 MORE INCIDENCE THEOREMS 144 132 QUADRILATERALS 149 133 EXERCISES 152 GRAFFITI 153 Measuring Angles and the Protractor Postulate 141 AXIOM 4 THE PROTRACTOR POSTULATE 155 142 PECULIAR PROTRACTORS 166 143 EXERCISES 169 Alternative Axiom Systems 151 HILBERTS AXIOMS 172 152 PIERIS POSTULATES 175 153 EXERCISES 180 Mirrors 161 RULERS AND PROTRACTORS 182 162 MIRROR AND SAS 184 163 EXERCISES 189 GRAFFITI 191 Congruence and the Penultimate Postulate 171 CONGRUENCE FOR TRIANGLES 192 SAS 195 173 CONGRUENCE THEOREMS 198 174 EXERCISES 201 GRAFFITI 202 Perpendiculars and Inequalities 181 A THEOREM ON PARALLELS 204 182 INEQUALITIES 207 183 RIGHT TRIANGLES 211 184 EXERCISES 213 GRAFFITI 215
 Absolute Geometry and Saccheri Quadrilaterals 211 EUCLIDS ABSOLUTE GEOMETRY 239 212 GIORDANOS THEOREM 248 213 EXERCISES 252 GRAFFITI 253 Saccheris Three Hypotheses 221 OMAR KHAYYAMS THEOREM 255 222 SACCHERIS THEOREM 260 223 EXERCISES 266 GRAFFITI 267 Euclids Parallel Postulate 231 EQUIVALENT STATEMENTS 269 232 INDEPENDENCE 281 233 EXERCISES 286 GRAFFITI 289 Biangles 241 CLOSED BIANGLES 292 242 CRITICAL ANGLES AND ABSOLUTE LENGTHS 295 243 THE INVENTION OF NONEUCLIDEAN GEOMETRY 302 244 EXERCISES 314 GRAFFITI 316 Excursions 251 PROSPECTUS 317 252 EUCLIDEAN GEOMETRY 320 253 HIGHER DIMENSIONS 323 254 EXERCISES 328 GRAFFITI 330 Parallels and the Ultimate Axiom 261 AXIOM 6 HPP 334 262 PARALLEL LINES 338 263 EXERCISES 344 GRAFFITI 346 Brushes and Cycles 271 BRUSHES 347 272 CYCLES 351 273 EXERCISES 356 GRAFFITI 358 Rotations Translations and Horolations 281 PRODUCTS OF TWO REFLECTIONS 360 282 REFLECTIONS IN LINES OF A BRUSH 365 283 EXERCISES 368 GRAFFITI 370 The Classification of Isometries 291 INVOLUTIONS 371 292 THE CLASSIFICATION THEOREM 378 293 EXERCISES 382 GRAFFITI 384 Symmetry 301 LEONARDOS THEOREM 386 302 FRIEZE PATTERNS 392 303 EXERCISES 397 GRAFFITI 400 Horocircles 311 LENGTH OF ARC 402 312 HYPERBOLIC FUNCTIONS 415 313 EXERCISES 417 GRAFFITI 419 The Fundamental Formula 321 TRIGONOMETRY 421 322 COMPLEMENTARY SEGMENTS 434 323 EXERCISES 439 GRAFFITI 443 Categoricalness and Area 331 ANALYTIC GEOMETRY 444 332 AREA 450 333 EXERCISES 459 GRAFFITI 463 Quadrature of the Circle 341 CLASSICAL THEOREMS 464 342 CALCULUS 474 343 CONSTRUCTIONS 479 344 EXERCISES 490 Hints and Answers 494 Notation Index 503 Index 504 حقوق النشر