Foundations of Geometry

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By Listen TheBook Posted on May 31, 2023

In Category - Mathematics

Authors: David Hilbert In Year : 1902

Book Language: English

Audiobook: Foundations of Geometry

Preface, Contents, and Introduction
The elements of geometry and the five groups of axioms
Group I: Axioms of connection
Group II: Axioms of Order
Consequences of the axioms of connection and order
Group III: Axioms of Parallels (Euclid's axiom)
Group IV: Axioms of congruence
Consequences of the axioms of congruence
Group V: Axiom of Continuity (Archimedes's axiom)
Compatibility of the axioms
Independence of the axioms of parallels. Non-euclidean geometry
Independence of the axioms of congruence
Independence of the axiom of continuity. Non-archimedean geometry
Complex number-systems
Demonstrations of Pascal's theorem
An algebra of segments, based upon Pascal's theorem
Proportion and the theorems of similitude
Equations of straight lines and of planes
Equal area and equal content of polygons
Parallelograms and triangles having equal bases and equal altitudes
The measure of area of triangles and polygons
Equality of content and the measure of area
Desargues's theorem and its demonstration for plane geometry by aid of the axiom of congruence
The impossibility of demonstrating Desargues's theorem for the plane with the help of the axioms of congruence
Introduction to the algebra of segments based upon the Desargues's theorme
The commutative and associative law of addition for our new algebra of segments
The associative law of multiplication and the two distributive laws for the new algebra of segments
Equation of straight line, based upon the new algebra of segments
The totality of segments, regarded as a complex number system
Construction of a geometry of space by aid of a desarguesian number system
Significance of Desargues's theorem
Two theorems concerning the possibility of proving Pascal's theorem
The commutative law of multiplication for an archimedean number system
The commutative law of multiplication for a non-archimedean number system
Proof of the two propositions concerning Pascal's theorem. Non-pascalian geometry
The demonstation, by means of the theorems of Pascal and Desargues
Analytic representation of the co-ordinates of points which can be so constructed
Geometrical constructions by means of a straight-edge and a transferer of segments
The representation of algebraic numbers and of integral rational functions as sums of squares
Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments
Conclusion
Appendix

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.