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| Preface | |
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| Introduction | |
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| Euclid's Geometry | |
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| Very Brief Survey of the Beginnings of Geometry | |
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| The Pythagoreans | |
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| Plato | |
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| Euclid of Alexandria | |
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| The Axiomatic Method | |
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| Undefined Terms | |
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| Euclid's First Four Postulates | |
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| The Parallel Postulate | |
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| Attempts to Prove the Parallel Postulate | |
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| The Danger in Diagrams | |
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| The Power of Diagrams | |
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| Straightedge-and-Compass Constructions, Briefly | |
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| Descartes' Analytic Geometry and Broader Idea of Constructions | |
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| Briefly on the Number [pi] | |
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| Conclusion | |
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| Logic and Incidence Geometry | |
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| Elementary Logic | |
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| Theorems and Proofs | |
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| RAA Proofs | |
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| Negation | |
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| Quantifiers | |
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| Implication | |
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| Law of Excluded Middle and Proof by Cases | |
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| Brief Historical Remarks | |
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| Incidence Geometry | |
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| Models | |
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| Consistency | |
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| Isomorphism of Models | |
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| Projective and Affine Planes | |
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| Brief History of Real Projective Geometry | |
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| Conclusion | |
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| Hilbert's Axioms | |
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| Flaws in Euclid | |
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| Axioms of Betweenness | |
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| Axioms of Congruence | |
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| Axioms of Continuity | |
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| Hilbert's Euclidean Axiom of Parallelism | |
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| Conclusion | |
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| Neutral Geometry | |
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| Geometry Without a Parallel Axiom | |
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| Alternate Interior Angle Theorem | |
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| Exterior Angle Theorem | |
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| Measure of Angles and Segments | |
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| Equivalence of Euclidean Parallel Postulates | |
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| Saccheri and Lambert Quadrilaterals | |
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| Angle Sum of a Triangle | |
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| Conclusion | |
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| History of the Parallel Postulate | |
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| Review | |
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| Proclus | |
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| Equidistance | |
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| Wallis | |
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| Saccheri | |
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| Clairaut's Axiom and Proclus' Theorem | |
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| Legendre | |
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| Lambert and Taurinus | |
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| Farkas Bolyai | |
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| The Discovery of Non-Euclidean Geometry | |
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| Janos Bolyai | |
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| Gauss | |
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| Lobachevsky | |
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| Subsequent Developments | |
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| Non-Euclidean Hilbert Planes | |
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| The Defect | |
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| Similar Triangles | |
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| Parallels Which Admit a Common Perpendicular | |
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| Limiting Parallel Rays, Hyperbolic Planes | |
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| Classification of Parallels | |
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| Strange New Universe? | |
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| Independence of the Parallel Postulate | |
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| Consistency of Hyperbolic Geometry | |
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| Beltrami's Interpretation | |
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| The Beltrami-Klein Model | |
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| The Poincare Models | |
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| Perpendicularity in the Beltrami-Klein Model | |
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| A Model of the Hyperbolic Plane from Physics | |
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| Inversion in Circles, Poincare Congruence | |
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| The Projective Nature of the Beltrami-Klein Model | |
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| Conclusion | |
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| Philosophical Implications, Fruitful Applications | |
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| What Is the Geometry of Physical Space? | |
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| What Is Mathematics About? | |
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| The Controversy about the Foundations of Mathematics | |
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| The Meaning | |
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| The Fruitfulness of Hyperbolic Geometry for Other Branches of Mathematics, Cosmology, and Art | |
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| Geometric Transformations | |
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| Klein's Erlanger Programme | |
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| Groups | |
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| Applications to Geometric Problems | |
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| Motions and Similarities | |
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| Reflections | |
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| Rotations | |
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| Translations | |
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| Half-Turns | |
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| Ideal Points in the Hyperbolic Plane | |
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| Parallel Displacements | |
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| Glides | |
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| Classification of Motions | |
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| Automorphisms of the Cartesian Model | |
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| Motions in the Poincare Model | |
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| Congruence Described by Motions | |
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| Symmetry | |
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| Further Results in Real Hyperbolic Geometry | |
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| Area and Defect | |
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| The Angle of Parallelism | |
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| Cycles | |
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| The Curvature of the Hyperbolic Plane | |
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| Hyperbolic Trigonometry | |
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| Circumference and Area of a Circle | |
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| Saccheri and Lambert Quadrilaterals | |
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| Coordinates in the Real Hyperbolic Plane | |
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| The Circumscribed Cycle of a Triangle | |
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| Bolyai's Constructions in the Hyperbolic Plane | |
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| Elliptic and Other Riemannian Geometries | |
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| Hilbert's Geometry Without Real Numbers | |
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| Axioms | |
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| Bibliography | |
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| Symbols | |
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| Name Index | |
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| Subject Index | |