Euclidean and Non-Euclidean Geometries: Development and History

Cover......Page 1
Preface......Page 11
Introduction......Page 23
Very Brief Survey of the Beginnings of Geometry......Page 29
The Pythagoreans......Page 31
Plato......Page 33
Euclid of Alexandria......Page 35
The Axiomatic Method......Page 37
Undefined Terms......Page 39
Euclid's First Four Postulates......Page 43
The Parallel Postulate......Page 48
Attempts to Prove the Parallel Postulate......Page 51
The Danger in Diagrams......Page 53
The Power of Diagrams......Page 55
Straightedge-and-Compass Constructions ,Briefly......Page 57
Descartes' Analytic Geometry and Broader Idea of Constructions......Page 62
Briefly on the Number π......Page 66
Conclusion......Page 68
Elementary Logic......Page 81
Theorems and Proofs......Page 83
RAA Proofs......Page 86
Negation......Page 88
Quantifiers......Page 89
Implication......Page 92
Law of Excluded Middle and Proof by Cases......Page 93
Brief Historical Remarks......Page 94
Incidence Geometry......Page 97
Models......Page 100
Consistency......Page 104
Isomorphism of Models......Page 107
Projective and Affine Planes......Page 109
Brief History of Real Projective Geometry......Page 117
Conclusion......Page 118
Flaws in Euclid......Page 131
Axioms of Betweenness......Page 133
Axioms of Congruence......Page 147
Axioms of Continuity......Page 157
Hilbert's Euclidean Axiom of Parallelism......Page 166
Conclusion......Page 170
Geometry Without a Parallel Axiom......Page 189
Alternate Interior Angle Theorem......Page 190
Exterior Angle Theorem......Page 192
Measure of Angles and Segments......Page 197
Equivalence of Euclidean Parallel Postulates......Page 201
Saccheri and Lambert Quadrilaterals......Page 204
Angle Sum of a Triangle......Page 211
Conclusion......Page 218
Review......Page 237
Proclus......Page 238
Equidistance......Page 241
Wallis......Page 242
Saccheri......Page 246
Clairaut's Axiom and Proclus' Theorem......Page 247
Legendre......Page 249
Lambert and Taurinus......Page 251
Farkas Bolyai......Page 253
Janos Bolya......Page 267
Gauss......Page 270
Lobachevsky......Page 273
Subsequent Development......Page 276
Non-Euclidean Hilbert Planes......Page 277
The Defect......Page 280
Similar Triangles......Page 281
Parallels Which Admit a Common Perpendicular......Page 282
Limiting Parallel Rays, Hyperbolic Planes......Page 285
Classification of Parallels......Page 290
Strange New Universe?......Page 292
Consistency of Hyperbolic Geometry......Page 317
Beltrami's Interpretation......Page 321
The Beltrami-Klein Model......Page 325
The Poincare Models......Page 330
Perpendicularity in the Beltrami-Klein Model......Page 336
A Model of the Hyperbolic Planefrom Physics......Page 339
Inversion in Circles, Poincare Congruence......Page 341
The Projective Nature of the Beltrami-KleinModel......Page 361
Conclusion......Page 374
What Is the Geometry of Physical Space?......Page 399
What Is Mathematics About?......Page 402
The Controversy about the Foundations ofMathematics......Page 404
The Meaning......Page 408
The Fruitfulness of Hyperbolic Geometry for Other Branches of Mathematics, Cosmology, and Art......Page 410
Klein's Erlanger Programme......Page 425
Groups......Page 427
Applications to Geometric Problems......Page 431
Motions and Similarities......Page 436
Reflections......Page 439
Rotations......Page 442
Translations......Page 445
Half-Turns......Page 448
Ideal Points in the Hyperbolic Plane......Page 450
Parallel Displacements......Page 452
Glides......Page 454
Classification of Motions......Page 455
Automorphisms of the Cartesian Model......Page 459
Motions in the Poincare Model......Page 464
Congruence Described by Motions......Page 472
Symmetry......Page 476
Chapter 10. Further Results in Real Hyperbolic Geometry......Page 503
Area and Defect......Page 504
The Angle of Parallelism......Page 508
Cycles......Page 509
The Curvature of the Hyperbolic Plane......Page 511
Hyperbolic Trigonometry......Page 515
Circumference and Area of a Circle......Page 524
Saccheri and Lambert Quadrilaterals......Page 528
Coordinates in the Real Hyperbolic Plane......Page 535
The Circumscribed Cycle of a Triangle......Page 543
Bolyai's Constructions in the Hyperbolic Plane......Page 548
Appendix A. Elliptic and Other Riemannian Geometries......Page 569
Appendix B. Hilbert’s Geometry Without Real Numbers......Page 599
Axioms......Page 625
Bibliography......Page 631
Symbols......Page 639
Name Index......Page 641
Subject Index......Page 645