Along with many small improvements, this revised edition contains van Yzeren's new proof of Pascal's theorem (§1.7) and, in Chapter 2, an improved treatment of order and sense. The Sylvester-Gallai theorem, instead of being introduced as a curiosity, is now used as an essential step in the theory of harmonic separation (§3.34). This makes the logi­ cal development self-contained: the footnotes involving the References (pp. 214-216) are for comparison with earlier treatments, and to give credit where it is due, not to fill gaps in the argument. H.S.M.C. November 1992 v Preface to the Second Edition Why should one study the real plane? To this question, put by those who advocate the complex plane, or geometry over a general field, I would reply that the real plane is an easy first step. Most of the prop­ erties are closely analogous, and the real field has the advantage of intuitive accessibility. Moreover, real geometry is exactly what is needed for the projective approach to non· Euclidean geometry. Instead of introducing the affine and Euclidean metrics as in Chapters 8 and 9, we could just as well take the locus of 'points at infinity' to be a conic, or replace the absolute involution by an absolute polarity.

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