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| Rules of the Road (Axiomatic Systems) | |
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| Historical Background: Axiomatic Systems and their Properties. Finite Geometries. Axioms for Incidence Geometry | |
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| Many Ways to Go | |
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| Introduction | |
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| Euclid's Geometry and Euclid's Elements | |
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| An Introduction to Modern Euclidean Geometries | |
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| Hilbert's Model for Euclidean Geometry | |
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| Birkhoff's Model for Euclidean Geometry | |
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| SMSG Postulates for Euclidean Geometry | |
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| Non-Euclidean Geometries | |
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| Traveling Together (Neutral Geometry) | |
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| Introduction | |
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| Preliminary Notions | |
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| Congruence Conditions | |
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| The Place of Parallels | |
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| The Saccheri-Legendre Theorem | |
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| The Search for a Rectangle | |
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| Summary | |
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| One Way to Go (Euclidean Geometry of the Plane) | |
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| Introduction | |
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| The Parallel Postulate and Some Implications | |
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| Congruence and Area | |
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| Similarity | |
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| Euclidean Results Concerning Circles | |
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| Some Euclidean Results Concerning Triangles | |
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| More Euclidean Results Concerning Triangles | |
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| The Nine-Point Circle | |
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| Euclidean Constructions | |
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| Summary | |
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| Side Trips (Analytic and Transformational Geometry) | |
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| Introduction | |
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| Analytic Geometry | |
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| Transformational Geometry | |
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| Analytic Transformations | |
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| Inversion | |
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| Summary | |
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| Other Ways to Go (Non-Euclidean Geometries) | |
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| Introduction | |
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| A Return to Neutral Geometry: The Angle of Parallelism | |
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| The Hyperbolic Parallel Postulate | |
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| Hyperbolic Results Concerning Polygons | |
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| Area in Hyperbolic Geometry | |
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| Showing Consistency: A Model for Hyperbolic Geometry | |
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| Classifying Theorems | |
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| Elliptic Geometry: A Geometry With No Parallels? Geometry in the Real World | |
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| Summary | |
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| All Roads Lead To . . . Projective Geometry | |
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| Introduction | |
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| The Real Projective Plane | |
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| Duality | |
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| Perspectivity | |
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| The Theorem of Desargues | |
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| Projective Transformations | |
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| Summary | |
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| Euclid's Definitions and Postulates Book I | |
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| Hilbert's Axioms for Euclidean Plane Geometry | |
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| Birkhoff's Postulates for Euclidean Plane Geometry | |
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| The SMSG Postulates for Euclidean Geometry | |
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| Geometer's SketchPad Scripts for Poincare Model of Hyperbolic Geometry | |
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| Bibliography | |
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| Index | |