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| Prologue: Euclid's Elements | |
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| Geometry before Euclid | |
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| The logical structure of Euclid's Elements | |
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| The historical significance of Euclid's Elements | |
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| A look at Book I of the Elements | |
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| A critique of Euclid's Elements | |
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| Final observations about the Elements | |
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| Axiomatic Systems and Incidence Geometry | |
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| The structure of an axiomatic system | |
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| An example: Incidence geometry | |
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| The parallel postulates in incidence geometry | |
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| Axiomatic systems and the real world | |
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| Theorems, proofs, and logic | |
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| Some theorems from incidence geometry | |
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| Axioms for Plane Geometry | |
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| The undefined terms and two fundamental axioms | |
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| Distance and the Ruler Postulate | |
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| Plane separation | |
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| Angle measure and the Protractor Postulate | |
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| The Crossbar Theorem and the Linear Pair Theorem | |
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| The Side-Angle-Side Postulate | |
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| The parallel postulates and models | |
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| Neutral Geometry | |
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| The Exterior Angle Theorem and perpendiculars | |
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| Triangle congruence conditions | |
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| Three inequalities for triangles | |
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| The Alternate Interior Angles Theorem | |
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| The Saccheri-Legendre Theorem | |
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| Quadrilaterals | |
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| Statements equivalent to the Euclidean Parallel Postulate | |
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| Rectangles and defect | |
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| The Universal Hyperbolic Theorem | |
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| Euclidean Geometry | |
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| Basic theorems of Euclidean geometry | |
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| The Parallel Projection Theorem | |
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| Similar triangles | |
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| The Pythagorean Theorem | |
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| Trigonometry | |
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| Exploring the Euclidean geometry of the triangle | |
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| Hyperbolic Geometry | |
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| The discovery of hyperbolic geometry | |
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| Basic theorems of hyperbolic geometry | |
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| Common perpendiculars | |
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| Limiting parallel rays and asymptotically parallel lines | |
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| Properties of the critical function | |
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| The defect of a triangle | |
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| Is the real world hyperbolic? | |
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| Area | |
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| The Neutral Area Postulate | |
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| Area in Euclidean geometry | |
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| Dissection theory in neutral geometry | |
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| Dissection theory in Euclidean geometry | |
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| Area and defect in hyperbolic geometry | |
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| Circles | |
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| Basic definitions | |
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| Circles and lines | |
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| Circles and triangles | |
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| Circles in Euclidean geometry | |
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| Circular continuity | |
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| Circumference and area of Euclidean circles | |
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| Exploring Euclidean circles | |
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| Constructions | |
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| Compass and straightedge constructions | |
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| Neutral constructions | |
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| Euclidean constructions | |
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| Construction of regular polygons | |
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| Area constructions | |
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| Three impossible constructions | |
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| Transformations | |
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| The transformational perspective | |
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| Properties of isometries | |
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| Rotations, translations, and glide reflections | |
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| Classification of Euclidean motions | |
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| Classification of hyperbolic motions | |
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| Similarity transformations in Euclidean geometry | |
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| A transformational approach to the foundations | |
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| Euclidean inversions in circles | |
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| Models | |
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| The significance of models for hyperbolic geometry | |
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| The Cartesian model for Euclidean geometry | |
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| The Poincar� disk model for hyperbolic geometry | |
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| Other models for hyperbolic geometry | |
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| Models for elliptic geometry | |
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| Regular Tessellations | |
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| Polygonal Models and the Geometry of Space | |
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| Curved surfaces | |
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| Approximate models for the hyperbolic plane | |
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| Geometric surfaces | |
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| The geometry of the universe | |
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| Conclusion | |
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| Further study | |
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| Templates | |
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| Appendices | |
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| Euclid's Book I | |
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| Definitions | |
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| Postulates | |
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| Common Notions | |
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| Propositions | |
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| Systems of Axioms for Geometry | |
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| Filling in Euclid's gaps | |
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| Hilbert's axioms | |
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| Birkhoff's axioms | |
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| MacLane's axioms | |
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| SMSG axioms | |
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| UCSMP axioms | |
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| The Postulates Used in this Book | |
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| The undefined terms | |
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| Neutral postulates | |
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| Parallel postulates | |
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| Area postulates | |
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| The reflection postulate | |
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| Logical relationships | |
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| Set Notation and the Real Numbers | |
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| Some elementary set theory | |
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| Properties of the real numbers | |
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| Functions | |
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| The van Hiele Model | |
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| Hints for Selected Exercises | |
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| Bibliography | |
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| Index | |