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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。 同世界知名教授H. S. M. Coxeter相比，没有哪个在世的几何学家可以把困难的题目写得更清晰、更优美。当非欧几何学*次被提出时，它似乎仅仅关乎与现实世界毫无关系的好奇心。而令所有人惊讶的是，它竟然对爱因斯坦广义相对论至关重要！Coxeter的书绝版太久了，向MAA再版这本经典著作脱帽致敬。 —Martin Gardner Coxeter的几何书籍是不应被丢失的珍品。我很高兴看到《非欧几何》重新出版。 —Doris Schattschneider

前辅文

I. THE HISTORICAL DEVELOPMENT OF NON-EUCLIDEAN GEOMETRY SECTION PAGE

1.1 Euclid

1.2 Saccheri and Lambert

1.3 Gauss, Wächter, 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 Conies

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 Ill

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 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

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 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 conformai model of the elliptic plane

14.7 Klein's conformai model of the hyperbolic plane

14.8 Poincaré's model of the hyperbolic plane

14.9 Conformai models of non-Euclidean space

XV. CONCLUDING REMARKS

15.1 HjelmsleVs 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

15.9 Inversive distance and the angle of parallelism

APPENDIX: ANGLES AND ARCS IN THE HYPERBOLIC PLANE

BIBLIOGRAPHY

INDEX

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书籍介绍

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。 同世界知名教授H. S. M. Coxeter相比，没有哪个在世的几何学家可以把困难的题目写得更清晰、更优美。当非欧几何学第一次被提出时，它似乎仅仅关乎与现实世界毫无关系的好奇心。而令所有人惊讶的是，它竟然对爱因斯坦广义相对论至关重要！Coxeter的书绝版太久了，向MAA再版这本经典著作脱帽致敬。 —Martin Gardner Coxeter的几何书籍是不应被丢失的珍品。我很高兴看到《非欧几何》重新出版。 —Doris Schattschneider