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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