导语

  

内容提要

  

    F.博斯克斯著的《几何三部曲(第1卷几何的公理化方法)(英文版)》以几何公理化方法的历史发展成果为基础，系统给出了欧几里得几何、非欧几里得几何和投影几何研究的现代方法。公理化几何是形式化数学的起源，其中有很多著名问题有待解决。对这些著名问题的研究往往会导致许多研究领域特别是代数研究领域的产生。基于公理化思想的数学理论是现代数学的基本特征。本书详尽地论述了公理化几何研究的内容，也给出了许多著名问题的完备性证明。

1 Pre-Heilenic Antiquity
  1.1 Prehistory
  1.2 Egypt
  1.3 Mesopotamia
  1.4 Problems
  1.5 Exercises
2 Some Pioneers of Greek Geometry
  2.1 Thales of Miletus
  2.2 Pythagoras and the Golden Ratio
  2.3 Trisecting the Angle
  2.4 Squaring the Circle
  2.5 Duplicating the Cube
  2.6 Incommensurable Magnitudes
  2.7 The Method of Exhaustion
  2.8 On the Continuity of Space
  2.9 Problems
  2.10 Exercises
3 Euclid's Elements
  3.1 Book 1: Straight Lines
  3.2 Book 2: Geometric Algebra
  3.3 Book 3: Circles
  3.4 Book 4: Polygons
  3.5 Book 5: Ratios
  3.6 Book 6: Similarities
  3.7 Book 7: Divisibility in Arithmetic
  3.8 Book 8: Geometric Progressions
  3.9 Book 9: More on Numbers
  3.10 Book 10: Incommensurable Magnitudes
  3.11 Book 11: Solid Geometry
  3.12 Book 12: The Method of Exhaustion
  3.13 Book 13: Regular Polyhedrons
  3.14 Problems
  3.15 Exercises
4 Some Masters of Greek Geometry
  4.1 Archimedes on the Circle
  4.2 Archimedes on the Number π
  4.3 Archimedes on the Sphere
  4.4 Archimedes on the Parabola
  4.5 Archimedes on the Spiral
  4.6 Apollonius on Conical Sections
  4.7 Apollonius on Conjugate Directions
  4.8 Apollonius on Tangents
  4.9 Apollonius on Poles and Polar Lines
  4.10 Apollonius on Foci
  4.11 Heron on the Triangle
  4.12 Menelaus on Trigonometry
  4.13 Ptolemy on Trigonometry
  4.14 Pappus on Anharmonic Ratios
  4.15 Problems
  4.16 Exercises
5 Post-Hellenic Euclidean Geometry
  5.1 Still Chasing the Number π
  5.2 The Medians of a Triangle
  5.3 The Altitudes of a Triangle
  5.4 Ceva's Theorem
  5.5 The Trisectrices of a Triangle
  5.6 Another Look at the Foci of Conics
  5.7 Inversions in the Plane
  5.8 Inversions in Solid Space
  5.9 The Stereographic Projection
  5.10 Let us Burn our Rulers!
  5.11 Problems
  5.12 Exercises
6 Projective Geometry
  6.1 Perspective Representation
  6.2 Projective Versus Euclidean
  6.3 Anharmonic Ratio
  6.4 The Desargues and the Pappus Theorems
  6.5 Axiomatic Projective Geometry
  6.6 Arguesian and Pappian Planes
  6.7 The Projective Plane over a Skew Field
  6.8 The Hilbert Theorems
  6.9 Problems
  6.10 Exercises
7 Non-Euclidean Geometry
  7.1 Chasing Euclid's Fifth Postulate
  7.2 The Saccheri Quadrilaterals
  7.3 The Angles of a Triangle
  7.4 The Limit Parallels
  7.5 The Area of a Triangle
  7.6 The Beltrami-Klein and Poincare Disks
  7.7 Problems
  7.8 Exercises
8 Hilbert's Axiomatization of the Plane
  8.1 The Axioms of Incidence
  8.2 The Axioms of Order
  8.3 The Axioms of Congruence
  8.4 The Axiom of Continuity
  8.5 The Axioms of Parallelism
  8.6 Problems
  8.7 Exercises
Appendix A Construetibility
  A.1 The Minimal Polynomial
  A.2 The Eisenstein Criterion
  A.3 Ruler and Compass Constructibility
  A.4 Constructibility Versus Field Theory
Appendix B The Classical Problems
  B. 1 Duplicating the Cube
  B.2 Trisecting the Angle
  B.3 Squaring the Circle
Appendix C Regular Polygons
  C.1 What the Greek Geometers Knew
  C.2 The Problem in Algebraic Terms
  C.3 Fermat Primes
  C.4 Elements of Modular Arithmetic
  C.5 A Flavour of Galois Theory
  C.6 The Gauss-Wantzel Theorem
References and Further Reading
Index