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Title: The Foundations of Geometry

Author: David Hilbert

Release Date: December 23, 2005 [eBook #17384]

[Most recently updated: June 13, 2022]

[Most recently updated: June 13, 2022]

Language: English

## Table of Links

**Introduction**

**Chapter I - THE FIVE GROUPS OF AXIOMS.**

- 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: Axiom of Parallels (Euclid’s axiom)
- Group IV: Axioms of congruence
- Consequences of the axioms of congruence
- Group V: Axiom of Continuity (Archimedes’s axiom)

**CHAPTER II - THE COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS.**

- 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

**CHAPTER III - THE THEORY OF PROPORTION.**

- Complex number-systems
- Demonstration 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

**CHAPTER IV - THE THEORY OF PLANE AREAS.**

- 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

**CHAPTER V - DESARGUES’S THEOREM.**

- Desargues’s theorem and its demonstration for plane geometry by aid of the axioms of congruence
- The impossibility of demonstrating Desargues’s theorem for the plane without the help of the axioms of congruence
- Introduction of an algebra of segments based upon Desargues’s theorem and independent of the axioms of congruence
- The commutative and the associative law of addition for our new algebra of segments
- The associative law of multiplication and the two distributive for the new algebra of segments
- Equation of the 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

**CHAPTER VI - PASCAL’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 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.**

- Geometrical constructions by means of a straight-edge and a

transferer of segments - Analytical representation of the co-ordinates of points which can be so constructed
- 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

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