《几何基础》是数学大师希尔伯特的一部名著，首次发表于1899年，该书第一次给出了完备的欧几里得几何公理系统。全体公理按性质分为五组（即关联公理、次序公理、合同公理、平行公理和连续公理），他对它们之间的逻辑关系作了深刻的考察，精确地提出了公理系统的相容性、独立性与完备性要求。为解决独立性问题，他的典型方法是构作一个模型，不满足所论的公理，但却满足所有其他公理。采用这种途径可赋予非欧几何以严密的逻辑解释，同时开拓了建立其他新几何学的可能性。对于相容性问题，他的重大贡献是借助于解析几何而将欧氏几何的相容性归结为初等算术的相容性。上述工作的意义远超出了几何基础的范围，而使他成为现代公理化方法的奠基人。

前辅文
INTRODUCTION
CHAPTER I THE FIVE GROUPS OF AXIOMS
  §1. The elements of geometry and the five groups of axioms
  §2. Group I. Axioms of connection
  §3. Group II. Axioms of order
  §4. Consequences of the axioms of connection and order
  §5. Group III. Axiom of parallels (Euclid’s axiom)
  §6. Group IV. Axioms of congruence
  §7. Consequences of the axioms of congruence
  §8. Group V. Axiom of continuity (Archimedes’s axiom)
CHAPTER II COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS
  §9. Compatibility of the axioms
  §10. Independence of the axioms of parallels (Non-euclidean geometry)
  §11. Independence of the axioms of congruence
  §12. Independence of the axiom of continuity (Non-archimedean geometry)
CHAPTER III THE THEORY OF PROPORTION
  §13. Complex number systems
  §14. Demonstration of Pascal’s theorem
  §15. An algebra of segments, based upon Pascal’s theorem
  §16. Proportion and the theorems of similitude
  §17. Equations of straight lines and of planes
CHAPTER IV THE THEORY OF PLANE AREAS
  §18. Equal area and equal content of polygons
  §19. Parallelograms and triangles having equal bases and equal altitudes
  §20. The measure of area of triangles and polygons
  §21. Equality of content and the measure of area
CHAPTER V DESARGUES’S THEOREM
  §22. Desargues’s theorem and its demonstration for plane geometry by aid of the axioms of congruence
  §23. The impossibility of demonstrating Desargues’s theorem for the plane without the help of the axioms of congruence
  §24. Introduction of an algebra of segments based upon Desargues’s theorem and independent of the axioms of congruence
  §25. The commutative and the associative law of addition for our new algebra of segments
  §26. The associative law of multiplication and the two distributive laws for the new algebra of segments
  §27. Equation of the straight line, based upon the new algebra of segments
  §28. The totality of segments, regarded as a complex number system
  §29. Construction of a geometry of space by aid of a desarguesian number system
  §30. Significance of Desargues’s theorem
CHAPTER VI PASCAL’S THEOREM
  §31. Two theorems concerning the possibility of proving Pascal’s theorem
  §32. The commutative law of multiplication for an Archimedean number system
  §33. The commutative law of multiplication for a non-archimedean number system
  §34. Proof of the two propositions concerning Pascal’s theorem (Non-pascalian geometry)
  §35. The demonstration, by means of the theorems of Pascal and Desargues, of any theorem relating to points of intersection
CHAPTER VII GEOMETRICAL CONSTRUCTIONS BASED UPON THE AXIOMS I–V
  §36. Geometrical constructions by means of a straight-edge and a transferer of segments
  §37. Analytical representation of the co-ordinates of points which can be so constructed
  §38. The representation of algebraic numbers and of integral rational functions as sums of squares
  §39. Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments
CONCLUSION
APPENDIX