Chapter I. The Attempts to prove Euclid's Parallel Postulate. |

1-5. The Greek Geometers and the Parallel Postulate |

6. The Arabs and the Parallel Postulate |

7-10. The Parallel Postulate during the Renaissance and the 17th Century |

Chapter II. The Forerunners on Non-Euclidean Geometry. |

11-17. GEROLAMO SACCHERI (1667-1733) |

18-22. JOHANN HEINRICH LAMBERT (1728-1777) |

23-26. The French Geometers towards the End of the 18th Century |

27-28. ADRIEN MARIE LEGENDRE (1752-1833) |

29. WOLFGANG BOLYAI (1775-1856) |

30. FRIEDRICH LUDWIG WACHTER (1792-1817) |

30. (bis) BERNHARD FRIEDRICH THIBAUT (1776-1832) |

Chapter III. The Founders of Non-Euclidean Geometry. |

31-34. KARL FRIEDRICH GAUSS (1777-1855) |

35. FERDINAND KARL SCHWEIKART (1780-1859) |

36-38. FRANZ ADOLF TAURINUS (1794-1874) |

Chapter IV. The Founders of Non-Euclidean Geometry (Cont.). |

39-45. NICOLAI IVANOVITSCH LOBATSCHEWSKY (1793-1856) |

46-55. JOHANN BOLYAI (1802-1860) |

56-58. The Absolute Trigonometry |

59. Hypotheses equivalent to Euclid's Postulate |

60-65. The Spread of Non-Euclidean Geometry |

Chapter V. The Later Development of Non-Euclidean Geometry. |

66. Introduction |

Differential Geometry and Non-Euclidean Geometry |

67-69. Geometry upon a Surface |

70-76. Principles of Plane Geometry on the Ideas of RIEMANN |

77. Principles of RIEMANN'S Solid Geometry |

78. The Work of HELMHOLTZ and the Investigations of LIE |

Projective Geometry and Non-Euclidean Geometry |

79-83. Subordination of Metrical Geometry to Projective Geometry |

84-91. Representation of the Geometry of LOBATSCHEWSKY-BOLYAI on the Euclidean Plane |

92. Representation of RIEMANN'S Elliptic Geometry in Euclidean Space |

93. Foundation of Geometry upon Descriptive Properties |

94. The Impossibility of proving Euclid's Postulate |

Appendix I. The Fundamental Principles of Statistics and Euclid's Postulate. |

1-3. On the Principle of the Lever |

4-8. On the Composition of Forces acting at a Point |

9-10. Non-Euclidean Statics |

11-12. Deduction of Plane Trigonometry from Statics |

Appendix II. CLIFFORD'S Parallels and Surface. Sketch of CLIFFFORD-KLEIN'S Problems. |

1-4. CLIFFORD'S Parallels |

5-8. CLIFFORD'S Surface |

9-11. Sketch of CLIFFORD-KLEIN'S Problem |

Appendix III. The Non-Euclidean Parallel Construction and other Allied Constructions. |

1-3. The Non-Euclidean Parallel Construction |

4. Construction of the Common Perpendicular to two non-intersecting Straight Lines |

5. Construction of the Common Parallel to the Straight Lines which bound an Angle |

6. Construction of the Straight Line which is perpendicular to one of the lines bounding an acute Angle and Parallel to the other |

7. The Absolute and the Parallel Construction |

Appendix IV. The Independence of Projective Geometry from Euclid's Postu |

1. Statement of the Problem |

2. Improper Points and the Complete Projective Plane |

3. The Complete Projective Line |

4. Combination of Elements |

5. Improper Lines |

6. Complete Projective Space |

7. Indirect Proof of the Independence of Projective Geometry from the Fifth Postulate |

8. BELTRAMI'S Direct Proof of this Independence |

Appendix V. The Impossibility of proving Euclid's Postulate. An Elementary Demonstration of this Impossibility founded upon the Properties of the System of Circles orthogonal to a Fixed Circle. |

1. Introduction |

2-7. The System of Circles passing through a Fixed Point |

8-12. The System of Circles orthogonal to a Fixed Circle |

Index of Authors |

The Science of Absolute Space and the Theory of Parallels |