Non-Euclidean Geometry (eBook)

Chapter I. The Attempts to prove Euclid's Parallel Postulate.1-5. The Greek Geometers and the Parallel Postulate6. The Arabs and the Parallel Postulate7-10. The Parallel Postulate during the Renaissance and the 17th CenturyChapter 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 Century27-28. 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 Trigonometry59. Hypotheses equivalent to Euclid's Postulate60-65. The Spread of Non-Euclidean GeometryChapter V. The Later Development of Non-Euclidean Geometry.66. Introduction Differential Geometry and Non-Euclidean Geometry67-69. Geometry upon a Surface70-76. Principles of Plane Geometry on the Ideas of RIEMANN77. Principles of RIEMANN'S Solid Geometry78. The Work of HELMHOLTZ and the Investigations of LIE Projective Geometry and Non-Euclidean Geometry79-83. Subordination of Metrical Geometry to Projective Geometry84-91. Representation of the Geometry of LOBATSCHEWSKY-BOLYAI on the Euclidean Plane92. Representation of RIEMANN'S Elliptic Geometry in Euclidean Space93. Foundation of Geometry upon Descriptive Properties94. The Impossibility of proving Euclid's PostulateAppendix I. The Fundamental Principles of Statistics and Euclid's Postulate.1-3. On the Principle of the Lever4-8. On the Composition of Forces acting at a Point9-10. Non-Euclidean Statics11-12. Deduction of Plane Trigonometry from StaticsAppendix II. CLIFFORD'S Parallels and Surface. Sketch of CLIFFFORD-KLEIN'S Problems.1-4. CLIFFORD'S Parallels5-8. CLIFFORD'S Surface9-11. Sketch of CLIFFORD-KLEIN'S ProblemAppendix III. The Non-Euclidean Parallel Construction and other Allied Constructions.1-3. The Non-Euclidean Parallel Construction4. Construction of the Common Perpendicular to two non-intersecting Straight Lines5. Construction of the Common Parallel to the Straight Lines which bound an Angle6. Construction of the Straight Line which is perpendicular to one of the lines bounding an acute Angle and Parallel to the other7. The Absolute and the Parallel ConstructionAppendix IV. The Independence of Projective Geometry from Euclid's Postulate.1. Statement of the Problem2. Improper Points and the Complete Projective Plane3. The Complete Projective Line4. Combination of Elements5. Improper Lines6. Complete Projective Space7. Indirect Proof of the Independence of Projective Geometry from the Fifth Postulate8. BELTRAMI'S Direct Proof of this IndependenceAppendix 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. Introduction2-7. The System of Circles passing through a Fixed Point8-12. The System of Circles orthogonal to a Fixed CircleIndex of AuthorsThe Science of Absolute Space and the Theory of Parallels___________________follow