Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein

PrefaceChapter I Axioms § 1. Axioms of Incidence § 2. Axioms of Betweenness § 3. Axiom of Continuity § 4. Axioms of MotionChapter II Consequences of the System of Axioms I § 5. Simple Properties of Straight Lines and Planes § 6. Desargues Configurations § 7. Linear Subspaces § 8. The Lattice of Linear Subspaces § 9. Basic Projective Configurations § 10. Projection and IntersectionChapter III Simple Consequences of the Systems of Axioms I—II § 11. Segments. Triangles § 12. Properties of Segments § 13. Linear Ordering § 14. Properties of Triangles § 15. The Tetrahedron § 16. Neighbourhoods § 17. Validity of the Systems of Axioms I—II for the Basic Domain ℛ § 18. Generalization of the Notion of Space § 19. The Extension and Restriction of SpacesChapter IV Projective Closure § 20. Half-Subspaces § 21. Half-Pencils. Angles § 22. Some Properties of Pencils and Bundles § 23. Coplanar Desargues Configurations § 24. Improper Pencils of Lines § 25. Improper Bundles of Lines § 26. The Projective Closure ℛ of ℛ § 27. The Projective Axioms § 28. The General CaseChapter V Investigation of the Projective Space § 29. Preliminaries § 30. Theorem of Duality in Projective Space § 31. Collineations § 32. The Erlangen Programme § 33. Theorem of Duality of the Plane § 34. Perspectivities and Projectivities § 35. Central Collineations of the Plane § 36. Separation § 37. Cyclic Ordering § 38. Projective Segments and Angles § 39. Complete Quadrangles. Harmonic Points § 40. Preliminaries About Coordinate Systems § 41. Coordinates in Projective Scales § 42. Halving a Projective Scale § 43. Coordinates for Dyadic Sets of Points on a LineChapter VI Consequences of the Systems of Axioms I, II, III § 44. Preliminaries § 45. Theorem Concerning the Infinite Point § 46. Coordinates in an Affine Line § 47. Coordinates on the Basic Projective Configurations of the First Degree § 48. Point-Coordinates in an Affine Plane § 49. The Fundamental Theorem of Projective Geometry § 50. Point-Coordinates in an Affine Space § 51. Vectors § 52. Homogeneous Point- and Plane-Coordinates in Space. Point- and Line-Coordinates in a Plane § 53. Determination of All Collineations of the Space § 54. Determination of the Coordinate Transformations of Space § 55. Transformation of Projective Coordinates § 56. Cross Ratio § 57. Imaginary Points § 58. Fixed Elements of Projectivities § 59. Involutions § 60. Involutory Collineations of a PlaneChapter VII Consequences of the Systems of Axioms I, II, III, IV § 61. Extended Motions § 62. The Comparability of Segments § 63. Reflections and Rotations. Absolute Polar Plane § 64. Metric Scales. Infinite and Ultra-Infinite Points. Elliptic, Parabolic and Hyperbolic Geometries § 65. Absolute Involution of Points on a Proper Line § 66. Midpoint and Bisector § 67. The Lines Perpendicular to a Proper Plane § 68. Motions as Products of Reflections § 69. Polarities with Respect to Surfaces and Curves of the Second Order § 70. The Absolute Configuration in the Elliptic Case § 71. The Absolute Configuration in the Hyperbolic Case § 72. Characterization of Motions in the Non-Parabolic Case § 73. The Absolute Configuration and Characterization of Motions in the Parabolic Case § 74. Formulae of Motion of the Three Geometries § 75. The Consistency of the Three Geometries § 76. Measuring of Segments § 77. Measuring of Angles § 78. Applications to TrigonometryBibliographyIndex