The Foundations of Geometry and the Non-Euclidean Plane

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Springer Science & Business Media, 1975 - Mathematics - 509 pages
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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.
  

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Contents

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
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