| The Foundation of Euclidean Geometry | |
| Introduction | p. 1 |
| The Definitions | p. 2 |
| The Common Notions | p. 4 |
| The Postulates | p. 4 |
| Tacit Assumptions Made by Euclid. Superposition | p. 5 |
| The Infinitude of the Line | p. 6 |
| Pasch's Axiom | p. 9 |
| The Principle of Continuity | p. 10 |
| The Postulate System of Hilbert | p. 12 |
| The Fifth Postulate | |
| Introduction | p. 17 |
| Substitutes for the Fifth Postulate | p. 20 |
| Playfair's Axiom | p. 20 |
| The Angle-Sum of a Triangle | p. 21 |
| The Existence of Similar Figures | p. 23 |
| Equidistant Straight Lines | p. 25 |
| Other Substitutes | p. 25 |
| Attempts to Prove the Fifth Postulate | p. 26 |
| Ptolemy | p. 26 |
| Proclus | p. 27 |
| Nasiraddin | p. 28 |
| Wallis | p. 29 |
| Saccheri | p. 30 |
| Lambert | p. 33 |
| Legendre | p. 34 |
| Some Fallacies in Attempts to Prove the Postulate | p. 39 |
| The Rotation Proof | p. 40 |
| Comparison of Infinite Areas | p. 41 |
| The Discovery of Non-Euclidean Geometry | |
| Introduction | p. 44 |
| Gauss | p. 45 |
| Bolyai | p. 48 |
| Lobachewsky | p. 53 |
| Wachter, Schweikart and Taurinus | p. 56 |
| Riemann | p. 60 |
| Further Developments | p. 63 |
| Conclusion | p. 63 |
| Hyperbolic Plane Geometry | |
| Introduction | p. 65 |
| The Characteristic Postulate of Hyperbolic Geometry | p. 66 |
| Elementary Properties of Parallels | p. 68 |
| Ideal Points | p. 71 |
| Some Properties of an Important Figure | p. 72 |
| The Angle of Parallelism | p. 76 |
| The Saccheri Quadrilateral | p. 77 |
| The Lambert Quadrilateral | p. 79 |
| The Sum of the Angles of a Triangle | p. 81 |
| The Common Perpendicular of Two Non-Intersecting Lines | p. 84 |
| Ultra-Ideal Points | p. 85 |
| The Variation in the Distance between Two Lines | p. 86 |
| The Perpendicular Bisectors of the Sides of a Triangle | p. 90 |
| The Construction of the Parallels to a Line through a Point | p. 93 |
| The Construction of a Common Parallel to Two Intersecting Lines | p. 97 |
| The Construction of a Line Perpendicular to One of Two Intersecting Lines and Parallel to the Other | p. 99 |
| Units of Length and Angle | p. 100 |
| Associated Right Triangles | p. 101 |
| The Construction of a Triangle when Its Angles Are Given | p. 105 |
| The Absolute | p. 107 |
| Circles | p. 109 |
| Corresponding Points | p. 110 |
| Limiting Curves and Their Properties | p. 113 |
| Equidistant Curves and Their Properties | p. 117 |
| The Limiting Curve as Related to Circles and Equidistant Curves | p. 119 |
| Area | p. 120 |
| Equivalence of Polygons and Triangles | p. 122 |
| Measure of Area | p. 127 |
| The Triangle with Maximum Area | p. 128 |
| Hyperbolic Plane Trigonometry | |
| Introduction | p. 131 |
| The Ratio of Corresponding Arcs of Concentric Limiting Curves | p. 131 |
| Relations between the Parts of an Important Figure | p. 136 |
| A Coördinate System and another Important Figure | p. 138 |
| The Relations between Complementary Segments | p. 140 |
| Relations among the Parts of a Right Triangle | p. 142 |
| Relations among the Parts of the General Triangle | p. 145 |
| The Relation between a Segment and Its Angle of Parallelism | p. 148 |
| Simplified Formulas for the Right Triangle and the General Triangle | p. 152 |
| The Parameter | p. 153 |
| Applications of Calculus to the Solutions of Some Problems in Hyperbolic Geometry | |
| Introduction | p. 157 |
| The Differential of Arc in Cartesian Coördinates | p. 158 |
| The Differential of Arc in Polar Coördinates | p. 160 |
| The Circumference of a Circle and the Lengths of Arcs of Limiting Curve and Equidistant Curve | p. 161 |
| The Area of a Fundamental Figure | p. 163 |
| Limiting Curve Coördinates | p. 165 |
| The Element of Area | p. 166 |
| The Area of a Circle | p. 169 |
| The Area of a Lambert Quadrilateral | p. 169 |
| The Area of a Triangle | p. 171 |
| Elliptic Plane Geometry and Trigonometry | |
| Introduction | p. 173 |
| The Characteristic Postulate of Elliptic Geometry and Its Immediate Consequences | p. 174 |
| The Relation between Geometry on a Sphere and Elliptic Geometry | p. 177 |
| The Two Elliptic Geometries | p. 179 |
| Properties of Certain Quadrilaterals | p. 180 |
| The Sum of the Angles of a Triangle | p. 182 |
| The Trigonometry of the Elliptic Plane | p. 185 |
| The Trigonometric Functions of an Angle | p. 185 |
| Properties of a Variable Lambert Quadrilateral | p. 190 |
| The Continuity of the Function (x) | p. 192 |
| An Important Functional Equation | p. 193 |
| The Function (x) | p. 194 |
| The Relation among the Parts of a Right Triangle | p. 196 |
| The Consistency of the Non-Euclidean Geometries | |
| Introduction | p. 201 |
| The Geometry of the Circles Orthogonal to a Fixed Circle | p. 204 |
| The Nominal Length of a Segment of Nominal Line | p. 206 |
| Displacement by Reflection | p. 208 |
| Displacement in the Geometry of the Nominal Lines | p. 209 |
| The Counterparts of Circles, Limiting Curves and Equidistant Curves | p. 213 |
| The Relation between a Nominal Distance and Its Angle of Parallelism | p. 214 |
| Conclusion | p. 217 |
| Appendix | |
| The Foundation of Euclidean Geometry | |
| The Definitions of Book I | p. 218 |
| The Postulates | p. 219 |
| The Common Notions | p. 220 |
| The Forty-Eight Propositions of Book I | p. 220 |
| Circular and Hyperbolic Functions | |
| The Trigonometric Functions | p. 223 |
| The Hyperbolic Functions | p. 225 |
| The Inverse Hyperbolic Functions | p. 228 |
| Geometric Interpretation of Circular and Hyperbolic Functions | p. 229 |
| The Theory of Orthogonal Circles and Allied Topics | |
| The Power of a Point with Regard to a Circle | p. 232 |
| The Radical Axis of Two Circles | p. 232 |
| Orthogonal Circles | p. 234 |
| Systems of Coaxal Circles | p. 235 |
| The Elements of Inversion | |
| Inversion | p. 237 |
| The Inverse of a Circle and the Inverse of a Line | p. 238 |
| The Effect of Inversion on Angles | p. 240 |
| The Peaucellier Inversor | p. 240 |
| Index | p. 245 |
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