H. S. M. Coxeter (1907–2003)

Indeholder "Preface", "Preface to the Third Edition", "Preface to the Fifth Edition", "I. The Historical Development of Non-Euclidean Geometry", " 1.1 Euclid", " 1.2 Saccheri and Lambert", " 1.3 Gauss, Wachter, Schweikart, Taurinus", " 1.4 Lobatschewsky", " 1.5 Bolyai", " 1.6 Riemann", " 1.7 Klein", "II. Real Projective Geometry: Foundations", " 2.1 Definitions and axioms", " 2.2 Models", " 2.3 The principle of duality", " 2.4 Harmonic sets", " 2.5 Sense", " 2.6 Triangular and tetrahedral regions", " 2.7 Ordered correspondences", " 2.8 One-dimensional projectivities", " 2.9 Involutions", "III. Real Projective Geometry, Polarities, Conics and Quadrics", " 3.1 Two-dimensional projectivities", " 3.2 Polarities in the plane", " 3.3 Conics", " 3.4 Projectivities on a conic", " 3.5 The fixed points of a collineation", " 3.6 Cones and reguli", " 3.7 Three-dimensional projectivities", " 3.8 Polarities in space", "IV. Homogenous coordinates", " 4.1 The von Staudt-Hessenberg calculus of points", " 4.2 One-dimensional projectivities", " 4.3 Coordinates in one and two dimensions", " 4.4 Collineations and coordinate transformations", " 4.5 Polarities", " 4.6 Coordinates in three dimensions", " 4.7 Three-dimensional projectivities", " 4.8 Line coordinates for the generators of a quatric", " 4.9 Complex projective geometry", "V. Elliptic Geometry in One Dimension", " 5.1 Elliptic geometry in general", " 5.2 Models", " 5.3 Reflections and translations", " 5.4 Congruence", " 5.5 Continuous translation", " 5.6 The length of a segment", " 5.7 Distance in terms of cross ratio", " 5.8 Alternative treatment using the complex line", "VI. Elliptic Geometry in Two Dimensions", " 6.1 Spherical and elliptic geometry", " 6.2 Reflection", " 6.3 Rotations and angles", " 6.4 Congruence", " 6.5 Circles", " 6.6 Composition of rotations", " 6.7 Formulae for distance and angle", " 6.8 Rotations and quaternions", " 6.9 Alternative treatment using the complex plane", "VII. Elliptic Geometry in Three Dimensions", " 7.1 Congruent transformations", " 7.2 Clifford parallels", " 7.3 The Stephanos-Cartan representation of rotations by points", " 7.4 Right translations and left translations", " 7.5 Right parallels and left parallels", " 7.6 Study's representation of lines by pairs of points", " 7.7 Clifford translations and quaternions", " 7.8 Study's coordinates for a line", " 7.9 Complex space", "VIII. Descriptive Geometry", " 8.1 Klein's projective model for hyperbolic geometry", " 8.2 Geometry in a convex region", " 8.3 Vebien's axioms of order", " 8.4 Order in a pencil", " 8.5 The geometry of lines and planes through a fixed point", " 8.6 Generalized bundles and pencils", " 8.7 Ideal points and lines", " 8.8 Verifying the projective axioms", " 8.9 Parallelism", "IX. Euclidean and Hyperbolic geometry", " 9.1 The introduction of congruence", " 9.2 Perpendicular lines and planes", " 9.3 Improper bundles and pencils", " 9.4 The absolute polarity", " 9.5 The Euclidean case", " 9.6 The hyperbolic case", " 9.7 The Absolute", " 9.8 The geometry of a bundle", "X. Hyperbolic geometry in Two Dimensions", " 10.1 Ideal elements", " 10.2 Angle-bisectors", " 10.3 Congruent transformations", " 10.4 Some famous constructions", " 10.5 An alternative expression for distance", " 10.6 The angle of parallelism", " 10.7 Distance and angle in terms of poles and polars", " 10.8 Canonical coordinates", " 10.9 Euclidean geometry as a limiting case", "XI. Circles and Triangles", " 11.1 Various definitions for a circle", " 11.2 The circle as a special conic", " 11.3 Spheres", " 11.4 The in- and ex-circles of a triangle", " 11.5 The circum-circles and centroids", " 11.6 The polar triangle and the orthocentre", "XII. The Use of A General Triangle of Reference", " 12.1 Formulae for distance and angle", " 12.2 The general circle", " 12.3 Tangential equations", " 12.4 Circum-circles and centroids", " 12.5 In- and ex-circles", " 12.6 The orthocentre", " 12.7 Elliptic trigonometry", " 12.8 The radii", " 12.9 Hyperbolic trigonometry", "XIII. Area", " 13.1 Equivalent regions", " 13.2 The choice of a unit", " 13.3 The area of a triangle in elliptic geometry", " 13.4 Area in hyperbolic geometry", " 13.5 The extension to three dimensions", " 13.6 The differential of distance", " 13.7 Area and areas of circles", " 13.8 Two surfaces which can be developed on the Euclidean plane", "XIV. Euclidean Models", " 14.1 The meaning of "elliptic" and "hyperbolic"", " 14.2 Beltrami's model", " 14.3 The differential of distance", " 14.4 Gnomonic projection", " 14.5 Development on surfaces of constant curvature", " 14.6 Klein's conformal model of the elliptic plane", " 14.7 Klein's conformal model of the hyperbolic plane", " 14.8 Poincaré's model of the hyperbolic plane", " 14.9 Conformal models of non-Euclidean space", "XV. Concluding Remarks", " 15.1 Hjelmslev's mid-line", " 15.2 The Napier chain", " 15.3 The Engel chain", " 15.4 Normalized canonical coordinates", " 15.5 Curvature", " 15.6 Quadratic forms", " 15.7 The volume of a tetrahedron", " 15.8 A brief historical survey of construction problems", "Bibliography", "Index".

Gennemgang af non-euklidisk geometri. Jeg synes ikke den er let at forstå. Fx fra side 261: "A flat pencil is represented by a system of coaxal circles (orthogonal to the fixed circle). In the case of a proper pencil, these will of course be intersecting circles. For a pencil of parallels, they will touch one another at a point on the fixed circle, their common tangent being a diameter." Måske skulle bogen bare have været fire gange så tyk og med flere illustrationer? Jeg har i alt fald brug for en lærer ved min side, der forstår alle spidsfindighederne, hvis jeg skal have en chance.… (meer)