The Foundations of Geometry

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"The material contained in the following translation was given in substance by Professor Hilbert as a course of lectures on euclidean geometry at the University of Göttingen during the winter semester of 1898–1899." Preface. Contents 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. (Archimedean 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. Chaper 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.