Euclidean and Non-Euclidean Geometries Development and History

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ISBN: 9780716799481 | 0716799480
Cover: Hardcover
Copyright: 9/28/2007

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This is the definitive presentation of the history, development and philosophical significance of non-Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Appropriate for liberal arts students, prospective high school teachers, math. majors, and even bright high school students. The first eight chapters are mostly accessible to any educated reader; the last two chapters and the two appendices contain more advanced material, such as the classification of motions, hyperbolic trigonometry, hyperbolic constructions, classification of Hilbert planes and an introduction to Riemannian geometry.

Preface	p. xiii
Introduction	p. xxv
Euclid's Geometry	p. 1
Very Brief Survey of the Beginnings of Geometry	p. 1
The Pythagoreans	p. 3
Plato	p. 5
Euclid of Alexandria	p. 7
The Axiomatic Method	p. 9
Undefined Terms	p. 11
Euclid's First Four Postulates	p. 15
The Parallel Postulate	p. 20
Attempts to Prove the Parallel Postulate	p. 23
The Danger in Diagrams	p. 25
The Power of Diagrams	p. 27
Straightedge-and-Compass Constructions, Briefly	p. 29
Descartes' Analytic Geometry and Broader Idea of Constructions	p. 34
Briefly on the Number [pi]	p. 38
Conclusion	p. 40
Logic and Incidence Geometry	p. 53
Elementary Logic	p. 53
Theorems and Proofs	p. 55
RAA Proofs	p. 58
Negation	p. 60
Quantifiers	p. 61
Implication	p. 64
Law of Excluded Middle and Proof by Cases	p. 65
Brief Historical Remarks	p. 66
Incidence Geometry	p. 69
Models	p. 72
Consistency	p. 76
Isomorphism of Models	p. 79
Projective and Affine Planes	p. 81
Brief History of Real Projective Geometry	p. 89
Conclusion	p. 90
Hilbert's Axioms	p. 103
Flaws in Euclid	p. 103
Axioms of Betweenness	p. 105
Axioms of Congruence	p. 119
Axioms of Continuity	p. 129
Hilbert's Euclidean Axiom of Parallelism	p. 138
Conclusion	p. 142
Neutral Geometry	p. 161
Geometry Without a Parallel Axiom	p. 161
Alternate Interior Angle Theorem	p. 162
Exterior Angle Theorem	p. 164
Measure of Angles and Segments	p. 169
Equivalence of Euclidean Parallel Postulates	p. 173
Saccheri and Lambert Quadrilaterals	p. 176
Angle Sum of a Triangle	p. 183
Conclusion	p. 190
History of the Parallel Postulate	p. 209
Review	p. 209
Proclus	p. 210
Equidistance	p. 213
Wallis	p. 214
Saccheri	p. 218
Clairaut's Axiom and Proclus' Theorem	p. 219
Legendre	p. 221
Lambert and Taurinus	p. 223
Farkas Bolyai	p. 225
The Discovery of Non-Euclidean Geometry	p. 239
Janos Bolyai	p. 239
Gauss	p. 242
Lobachevsky	p. 245
Subsequent Developments	p. 248
Non-Euclidean Hilbert Planes	p. 249
The Defect	p. 252
Similar Triangles	p. 253
Parallels Which Admit a Common Perpendicular	p. 254
Limiting Parallel Rays, Hyperbolic Planes	p. 257
Classification of Parallels	p. 262
Strange New Universe?	p. 264
Independence of the Parallel Postulate	p. 289
Consistency of Hyperbolic Geometry	p. 289
Beltrami's Interpretation	p. 293
The Beltrami-Klein Model	p. 297
The Poincare Models	p. 302
Perpendicularity in the Beltrami-Klein Model	p. 308
A Model of the Hyperbolic Plane from Physics	p. 311
Inversion in Circles, Poincare Congruence	p. 313
The Projective Nature of the Beltrami-Klein Model	p. 333
Conclusion	p. 346
Philosophical Implications, Fruitful Applications	p. 371
What Is the Geometry of Physical Space?	p. 371
What Is Mathematics About?	p. 374
The Controversy about the Foundations of Mathematics	p. 376
The Meaning	p. 380
The Fruitfulness of Hyperbolic Geometry for Other Branches of Mathematics, Cosmology, and Art	p. 382
Geometric Transformations	p. 397
Klein's Erlanger Programme	p. 397
Groups	p. 399
Applications to Geometric Problems	p. 403
Motions and Similarities	p. 408
Reflections	p. 411
Rotations	p. 414
Translations	p. 417
Half-Turns	p. 420
Ideal Points in the Hyperbolic Plane	p. 422
Parallel Displacements	p. 424
Glides	p. 426
Classification of Motions	p. 427
Automorphisms of the Cartesian Model	p. 431
Motions in the Poincare Model	p. 436
Congruence Described by Motions	p. 444
Symmetry	p. 448
Further Results in Real Hyperbolic Geometry	p. 475
Area and Defect	p. 476
The Angle of Parallelism	p. 480
Cycles	p. 481
The Curvature of the Hyperbolic Plane	p. 483
Hyperbolic Trigonometry	p. 487
Circumference and Area of a Circle	p. 496
Saccheri and Lambert Quadrilaterals	p. 500
Coordinates in the Real Hyperbolic Plane	p. 507
The Circumscribed Cycle of a Triangle	p. 515
Bolyai's Constructions in the Hyperbolic Plane	p. 520
Elliptic and Other Riemannian Geometries	p. 541
Hilbert's Geometry Without Real Numbers	p. 571
Axioms	p. 597
Bibliography	p. 603
Symbols	p. 611
Name Index	p. 613
Subject Index	p. 617
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Euclidean and Non-Euclidean Geometries Development and History

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