Foundations of Geometry
David Hilbert
Play
Length 5 h 27 min
Chapters 41
First published 1902
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.
Chapters
1
Preface, Contents, and Introduction Read by Jim Wrenholt 11:44
2
Group I: Axioms of connection Read by Jim Wrenholt 3:55
3
Group II: Axioms of Order Read by Jim Wrenholt 3:23
4
Consequences of the axioms of connection and order Read by Jim Wrenholt 7:00
5
Group III: Axioms of Parallels (Euclid's axiom) Read by Jim Wrenholt 2:33
6
Group IV: Axioms of congruence Read by Jim Wrenholt 8:38
7
Consequences of the axioms of congruence Read by Jim Wrenholt 20:38
8
Group V: Axiom of Continuity (Archimedes's axiom) Read by Jim Wrenholt 4:20
9
Compatibility of the axioms Read by Jim Wrenholt 6:36
10
Independence of the axioms of parallels. Non-euclidean geometry Read by Jim Wrenholt 4:59
11
Independence of the axioms of congruence Read by Jim Wrenholt 6:25
12
Independence of the axiom of continuity. Non-archimedean geometry Read by Jim Wrenholt 6:24
13
Complex number-systems Read by Jim Wrenholt 6:33
14
Demonstrations of Pascal's theorem Read by Jim Wrenholt 14:50
15
An algebra of segments, based upon Pascal's theorem Read by Jim Wrenholt 7:02
16
Proportion and the theorems of similitude Read by Jim Wrenholt 5:59
17
Equations of straight lines and of planes Read by Jim Wrenholt 7:49
18
Equal area and equal content of polygons Read by Jim Wrenholt 5:34
19
Parallelograms and triangles having equal bases and equal altitudes Read by Jim Wrenholt 5:52
20
The measure of area of triangles and polygons Read by Jim Wrenholt 10:05
21
Equality of content and the measure of area Read by Jim Wrenholt 8:01
22
Desargues's theorem and its demonstration for plane geometry by aid of the axiom of congruence Read by Jim Wrenholt 6:25
23
The impossibility of demonstrating Desargues's theorem for the plane with the help of the axioms of congruence Read by Jim Wrenholt 10:15
24
Introduction to the algebra of segments based upon the Desargues's theorme Read by Jim Wrenholt 4:58
25
The commutative and associative law of addition for our new algebra of segments Read by Jim Wrenholt 4:16
26
The associative law of multiplication and the two distributive laws for the new algebra of segments Read by Jim Wrenholt 12:16
27
Equation of straight line, based upon the new algebra of segments Read by Jim Wrenholt 8:17
28
The totality of segments, regarded as a complex number system Read by Jim Wrenholt 3:45
29
Construction of a geometry of space by aid of a desarguesian number system Read by Jim Wrenholt 9:05
30
Significance of Desargues's theorem Read by Jim Wrenholt 3:18
31
Two theorems concerning the possibility of proving Pascal's theorem Read by Jim Wrenholt 3:13
32
The commutative law of multiplication for an archimedean number system Read by Jim Wrenholt 5:23
33
The commutative law of multiplication for a non-archimedean number system Read by Jim Wrenholt 9:46
34
Proof of the two propositions concerning Pascal's theorem. Non-pascalian geometry Read by Jim Wrenholt 3:33
35
The demonstation, by means of the theorems of Pascal and Desargues Read by Jim Wrenholt 5:29
36
Analytic representation of the co-ordinates of points which can be so constructed Read by Jim Wrenholt 7:34
37
Geometrical constructions by means of a straight-edge and a transferer of segments Read by Jim Wrenholt 6:51
38
The representation of algebraic numbers and of integral rational functions as sums of squares Read by Jim Wrenholt 12:44
39
Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments Read by Jim Wrenholt 12:02
40
Conclusion Read by Jim Wrenholt 14:09
41
Appendix Read by Jim Wrenholt 22:31
From LibriVox, read by volunteers and in the public domain. LibriVox page (opens librivox.org) , the text (opens gutenberg.org) , all files on the Internet Archive (opens archive.org) .
