Foundations of Geometry

Author(s)

• David Hilbert

The German mathematician David Hilbert was one of the most influential mathematicians of the 19th/early 20th century. Hilbert's 20 axioms were first proposed by him in 1899 in his book Grundlagen der Geometrie as the foundation for a modern treatment of Euclidean geometry.

Hilbert's axiom system is constructed with six primitive notions: the three primitive terms point, line, and plane, and the three primitive relations Betweenness (a ternary relation linking points), Lies on (or Containment, three binary relations between the primitive terms), and Congruence (two binary relations, one linking line segments and one linking angles).

The original monograph in German was based on Hilbert's own lectures and was organized by himself for a memorial address given in 1899. This was quickly followed by a French translation with changes made by Hilbert; an authorized English translation was made by E.J. Townsend in 1902. This translation - from which this audiobook has been read - already incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition.

1. Preface, Contents, and Introduction
2. The elements of geometry and the five groups of axioms
3. Group I: Axioms of connection
4. Group II: Axioms of Order
5. Consequences of the axioms of connection and order
6. Group III: Axioms of Parallels (Euclid's axiom)
7. Group IV: Axioms of congruence
8. Consequences of the axioms of congruence
9. Group V: Axiom of Continuity (Archimedes's axiom)
10. Compatibility of the axioms
11. Independence of the axioms of parallels. Non-euclidean geometry
12. Independence of the axioms of congruence
13. Independence of the axiom of continuity. Non-archimedean geometry
14. Complex number-systems
15. Demonstrations of Pascal's theorem
16. An algebra of segments, based upon Pascal's theorem
17. Proportion and the theorems of similitude
18. Equations of straight lines and of planes
19. Equal area and equal content of polygons
20. Parallelograms and triangles having equal bases and equal altitudes
21. The measure of area of triangles and polygons
22. Equality of content and the measure of area
23. Desargues's theorem and its demonstration for plane geometry by aid of the axiom of congruence
24. The impossibility of demonstrating Desargues's theorem for the plane with the help of the axioms of congruence
25. Introduction to the algebra of segments based upon the Desargues's theorme
26. The commutative and associative law of addition for our new algebra of segments
27. The associative law of multiplication and the two distributive laws for the new algebra of segments
28. Equation of straight line, based upon the new algebra of segments
29. The totality of segments, regarded as a complex number system
30. Construction of a geometry of space by aid of a desarguesian number system
31. Significance of Desargues's theorem
32. Two theorems concerning the possibility of proving Pascal's theorem
33. The commutative law of multiplication for an archimedean number system
34. The commutative law of multiplication for a non-archimedean number system
35. Proof of the two propositions concerning Pascal's theorem. Non-pascalian geometry
36. The demonstation, by means of the theorems of Pascal and Desargues
37. Analytic representation of the co-ordinates of points which can be so constructed
38. Geometrical constructions by means of a straight-edge and a transferer of segments
39. The representation of algebraic numbers and of integral rational functions as sums of squares
40. Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments
41. Conclusion
42. Appendix