导语

  

内容提要

  

    贯穿本书大部分内容的二维或三维空间的非欧几何，被视为与一组简单公理相关的、实射影几何的特例，这组公理涉及点、线、面、关联、序和连续性，未涉及距离或角度的测量。综述之后，作者从Von Staudt的思想——将点视为可以相加或相乘的实体——出发，引入齐次坐标。保持关联的变换称为直射变换，它们自然地导出等距同构或“全等变换”。遵循Bertrand Russell的建议，连续性用序来描述。通过特殊化椭圆或双曲配极——将点变换为线（二维）、面（三维），反之亦然——椭圆和双曲几何可从实射影几何派生而来。
    本书的一个不同寻常的特点是，它利用一般的线性坐标变换，来推导椭圆和双曲三角函数的公式。根据Gauss的巧妙想法，三角形面积与其角度之和有关。
    任何熟悉代数乃至群论基础的读者都可以从本书获益。第六版澄清了第五版的一些晦涩之处，新增的15.9节包含了作者非常有用的反演距离的概念。

Ⅰ. THE HISTORICAL DEVELOPMENT OF NON-EUCLIDEAN GEOMETRY
  SECTION
  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 PROECTITVE GFOMETRYE 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
  SECTION
  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. HOMOGENEOUS 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 quadric
  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
SECTION
  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  Veblen'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
  SECTION
  9.5  The Euclidean case
  9.6  The hyperbolic case
  9.7  The Absolute
  9.8  The geometry of a bundle
Ⅹ. НYPERBOLIC GEOMETRY IN TWO DMENSIONS
  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 ter ms of poles and polars
  10.8  Canonical coordinates
  10.9  Euclidean geo metry 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
  SECTION
  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  Arcs 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 confor mal 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
  SECTION
  15.8  A brief historical survey of construction problems
  15.9  Inversive distance and the angle of parallelism
APPENDIX: ANGLES AND ARCS IN THE HYPERBOLIC PLANE
BIBLIOGRAPHY
INDEX