Description: 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 Generalized perspectivity.- 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.

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