The Real Projective Plane

1. A Comparison of Various Kinds of Geometry. - 11 Introduction. - 12 Parallel projection. - 13 Central projection. - 14 The line at infinity. - 15 Desargues's two-triangle theorem. - 16 The directed angle or cross. - 17 Hexagramma mysticum. - 18 An outline of subsequent work. - 2. Incidence. - 11 Primitive concepts. - 22 The axioms of incidence. - 23 The principle of duality. - 24 Quadrangle and quadrilateral. - 25 Harmonic conjugacy. - 26 Ranges and pencils. - 27 Perspectivity. - 28 The invariance and symmetry of the harmonic relation. - 3. Order and Continuity. - 31 The axioms of order. - 32 Segment and interval. - 33 Sense. - 34 Ordered correspondence. - 35 Continuity. - 36 Invariant points. - 37 Order in a pencil. - 38 The four regions determined by a triangle. - 4. One-Dimensional Projectivities. - 41 Projectivity. - 42 The fundamental theorem of projective geometry. - 43 Pappus's theorem. - 44 Classification of projectivities. - 45 Periodic projectivities. - 46 Involution. - 47 Quadrangular set of six points. - 48 Projective pencils. - 5. Two-Dimensional Projectivities. - 51 Collineation. - 52 Perspective collineation. - 53 Involutory collineation. - 54 Correlation. - 55 Polarity. - 56 Polar and self-polar triangles. - 57 The self-polarity of the Desargues configuration. - 58 Pencil and range of polarities. - 59 Degenerate polarities. - 6. Conics. - 61 Historial remarks. - 62 Elliptic and hyperbolic polarities. - 63 How a hyperbolic polarity determines a conic. - 64 Conjugate points and conjugate lines. - 65 Two possible definitions for a conic. - 66 Construction for the conic through five given points. - 67 Two triangles inscribed in a conic. - 68 Pencils of conics. - 7. Projectivities on a Conic. - 71 Generalizedperspectivity. - 72 Pascal and Brianchon. - 73 Construction for a projectivity on a conic. - 74 Construction for the invariant points of a given hyperbolic projectivity. - 75 Involution on a conic. - 76 A generalization of Steiner's construction. - 77 Trilinear polarity. - 8. Affine Geometry. - 81 Parallelism. - 82 Intermediacy. - 83 Congruence. - 84 Distance. - 85 Translation and dilatation. - 86 Area. - 87 Classification of conics. - 88 Conjugate diameters. - 89 Asymptotes. - 810 Affine transformations and the Erlangen programme. - 9. Euclidean Geometry. - 91 Perpendicularity. - 92 Circles. - 93 Axes of a conic. - 94 Congruent segments. - 95 Congruent angles. - 96 Congruent transformations. - 97 Foci. - 98 Directrices. - 10. Continuity. - 101 An improved axiom of continuity. - 102 Proving Archimedes' axiom. - 103 Proving the line to be perfect. - 104 The fundamental theorem of projective geometry. - 105 Proving Dedekind's axiom. - 106 Enriques's theorem. - 11. The Introduction of Coordinates. - 111 Addition of points. - 112 Multiplication of points. - 113 Rational points. - 114 Projectivities. - 115 The one-dimensional continuum. - 116 Homogeneous coordinates. - 117 Proof that a line has a linear equation. - 118 Line coordinates. - 12. The Use of Coordinates. - 121 Consistency and categoricalness. - 122 Analytic geometry. - 123 Verifying the axioms of incidence. - 124 Verifying the axioms of order and continuity. - 125 The general collineation. - 126 The general polarity. - 127 Conies. - 128 The affine plane: affine and areal coordinates. - 129 The Euclidean plane: Cartesian and trilinear coordinates. Language: English