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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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