Table of Contents

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 Postulate. 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 follow