The Real Projective Plane

1. A Comparison of Various Kinds of Geometry.- 1*1 Introduction.- 1*2 Parallel projection.- 1*3 Central projection.- 1*4 The line at infinity.- 1*5 Desargues's two-triangle theorem.- 1*6 The directed angle, or cross.- 1*7 Hexagramma mysticum.- 1*8 An outline of subsequent work.- 2. Incidence.- 1*1 Primitive concepts.- 2*2 The axioms of incidence.- 2*3 The principle of duality.- 2*4 Quadrangle and quadrilateral.- 2*5 Harmonic conjugacy.- 2*6 Ranges and pencils.- 2*7 Perspectivity.- 2*8 The invariance and symmetry of the harmonic relation.- 3. Order and Continuity.- 3*1 The axioms of order.- 3*2 Segment and interval.- 3*3 Sense.- 3*4 Ordered correspondence.- 3*5 Continuity.- 3*6 Invariant points.- 3*7 Order in a pencil.- 3*8 The four regions determined by a triangle.- 4. One-Dimensional Projectivities.- 4*1 Projectivity.- 4*2 The fundamental theorem of projective geometry.- 4*3 Pappus's theorem.- 4*4 Classification of projectivities.- 4*5 Periodic projectivities.- 4*6 Involution.- 4*7 Quadrangular set of six points.- 4*8 Projective pencils.- 5. Two-Dimensional Projectivities.- 5*1 Collineation.- 5*2 Perspective collineation.- 5*3 Involutory collineation.- 5*4 Correlation.- 5*5 Polarity.- 5*6 Polar and self-polar triangles.- 5*7 The self-polarity of the Desargues configuration.- 5*8 Pencil and range of polarities.- 5*9 Degenerate polarities.- 6. Conics.- 6*1 Historial remarks.- 6*2 Elliptic and hyperbolic polarities.- 6*3 How a hyperbolic polarity determines a conic.- 6*4 Conjugate points and conjugate lines.- 6*5 Two possible definitions for a conic.- 6*6 Construction for the conic through five given points.- 6*7 Two triangles inscribed in a conic.- 6*8 Pencils of conics.- 7. Projectivities on a Conic.- 7*1 Generalized perspectivity.- 7*2 Pascal and Brianchon.- 7*3 Construction for a projectivity on a conic.- 7*4 Construction for the invariant points of a given hyperbolic projectivity.- 7*5 Involution on a conic.- 7*6 A generalization of Steiner's construction.- 7*7 Trilinear polarity.- 8. Affine Geometry.- 8*1 Parallelism.- 8*2 Intermediacy.- 8*3 Congruence.- 8*4 Distance.- 8*5 Translation and dilatation.- 8*6 Area.- 8*7 Classification of conics.- 8*8 Conjugate diameters.- 8*9 Asymptotes.- 8*10 Affine transformations and the Erlangen programme.- 9. Euclidean Geometry.- 9*1 Perpendicularity.- 9*2 Circles.- 9*3 Axes of a conic.- 9*4 Congruent segments.- 9*5 Congruent angles.- 9*6 Congruent transformations.- 9*7 Foci.- 9*8 Directrices.- 10. Continuity.- 10*1 An improved axiom of continuity.- 10*2 Proving Archimedes' axiom.- 10*3 Proving the line to be perfect.- 10*4 The fundamental theorem of projective geometry.- 10*5 Proving Dedekind's axiom.- 10*6 Enriques's theorem.- 11. The Introduction of Coordinates.- 11*1 Addition of points.- 11*2 Multiplication of points.- 11*3 Rational points.- 11*4 Projectivities.- 11*5 The one-dimensional continuum.- 11*6 Homogeneous coordinates.- 11*7 Proof that a line has a linear equation.- 11*8 Line coordinates.- 12. The Use of Coordinates.- 12*1 Consistency and categoricalness.- 12*2 Analytic geometry.- 12*3 Verifying the axioms of incidence.- 12*4 Verifying the axioms of order and continuity.- 12*5 The general collineation.- 12*6 The general polarity.- 12*7 Conies.- 12*8 The affine plane: affine and areal coordinates.- 12*9 The Euclidean plane: Cartesian and trilinear coordinates.