Foundations of Geometry


Read by Jim Wrenholt

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. (5 hr 26 min)

Chapters

Preface, Contents, and Introduction 11:44 Read by Jim Wrenholt
The elements of geometry and the five groups of axioms 2:30 Read by Jim Wrenholt
Group I: Axioms of connection 3:55 Read by Jim Wrenholt
Group II: Axioms of Order 3:23 Read by Jim Wrenholt
Consequences of the axioms of connection and order 7:00 Read by Jim Wrenholt
Group III: Axioms of Parallels (Euclid's axiom) 2:33 Read by Jim Wrenholt
Group IV: Axioms of congruence 8:38 Read by Jim Wrenholt
Consequences of the axioms of congruence 20:38 Read by Jim Wrenholt
Group V: Axiom of Continuity (Archimedes's axiom) 4:20 Read by Jim Wrenholt
Compatibility of the axioms 6:36 Read by Jim Wrenholt
Independence of the axioms of parallels. Non-euclidean geometry 4:59 Read by Jim Wrenholt
Independence of the axioms of congruence 6:25 Read by Jim Wrenholt
Independence of the axiom of continuity. Non-archimedean geometry 6:24 Read by Jim Wrenholt
Complex number-systems 6:33 Read by Jim Wrenholt
Demonstrations of Pascal's theorem 14:50 Read by Jim Wrenholt
An algebra of segments, based upon Pascal's theorem 7:02 Read by Jim Wrenholt
Proportion and the theorems of similitude 5:59 Read by Jim Wrenholt
Equations of straight lines and of planes 7:49 Read by Jim Wrenholt
Equal area and equal content of polygons 5:34 Read by Jim Wrenholt
Parallelograms and triangles having equal bases and equal altitudes 5:52 Read by Jim Wrenholt
The measure of area of triangles and polygons 10:05 Read by Jim Wrenholt
Equality of content and the measure of area 8:01 Read by Jim Wrenholt
Desargues's theorem and its demonstration for plane geometry by aid of the axio… 6:25 Read by Jim Wrenholt
The impossibility of demonstrating Desargues's theorem for the plane with the h… 10:15 Read by Jim Wrenholt
Introduction to the algebra of segments based upon the Desargues's theorme 4:58 Read by Jim Wrenholt
The commutative and associative law of addition for our new algebra of segments 4:16 Read by Jim Wrenholt
The associative law of multiplication and the two distributive laws for the new… 12:16 Read by Jim Wrenholt
Equation of straight line, based upon the new algebra of segments 8:17 Read by Jim Wrenholt
The totality of segments, regarded as a complex number system 3:45 Read by Jim Wrenholt
Construction of a geometry of space by aid of a desarguesian number system 9:05 Read by Jim Wrenholt
Significance of Desargues's theorem 3:18 Read by Jim Wrenholt
Two theorems concerning the possibility of proving Pascal's theorem 3:13 Read by Jim Wrenholt
The commutative law of multiplication for an archimedean number system 5:23 Read by Jim Wrenholt
The commutative law of multiplication for a non-archimedean number system 9:46 Read by Jim Wrenholt
Proof of the two propositions concerning Pascal's theorem. Non-pascalian geomet… 3:33 Read by Jim Wrenholt
The demonstation, by means of the theorems of Pascal and Desargues 5:29 Read by Jim Wrenholt
Analytic representation of the co-ordinates of points which can be so construct… 7:34 Read by Jim Wrenholt
Geometrical constructions by means of a straight-edge and a transferer of segme… 6:51 Read by Jim Wrenholt
The representation of algebraic numbers and of integral rational functions as s… 12:44 Read by Jim Wrenholt
Criterion for the possibility of a geometrical construction by means of a strai… 12:02 Read by Jim Wrenholt
Conclusion 14:09 Read by Jim Wrenholt
Appendix 22:31 Read by Jim Wrenholt