Book:David Hilbert/The Foundations of Geometry
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David Hilbert: Foundations of Geometry
Published $\text {1902}$ (translated by E.J. Townsend)
Subject Matter
Contents
- Introduction
- CHAPTER $\text I$. THE FIVE GROUPS OF AXIOMS.
- $\S 1$. The elements of geometry and the five groups of axioms
- $\S 2$. Group $\text I$: Axioms of connection
- $\S 3$. Group $\text {II}$: Axioms of Order
- $\S 4$. Consequences of the axioms of connection and order
- $\S 5$. Group $\text {III}$: Axiom of Parallels (Euclid's axiom)
- $\S 6$. Group $\text {IV}$: Axioms of congruence
- $\S 7$. Consequences of the axioms of congruence
- $\S 8$. Group $\text V$: Axiom of Continuity (Archimedes's axiom)
- CHAPTER $\text {II}$. THE COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS.
- $\S 9$. Compatibility of the axioms
- $\S 10$. Independence of the axioms of parallels. Non-euclidean geometry
- $\S 11$. Independence of the axioms of congruence
- $\S 12$. Independence of the axiom of continuity. Non-archimedean geometry
- CHAPTER $\text {III}$. THE THEORY OF PROPORTION.
- $\S 13$. Complex number-systems
- $\S 14$. Demonstration of Pascal's theorem
- $\S 15$. An algebra of segments, based upon Pascal's theorem
- $\S 16$. Proportion and the theorems of similitude
- $\S 17$. Equations of straight lines and of planes
- CHAPTER $\text {IV}$. THE THEORY OF PLANE AREAS.
- $\S 18$. Equal area and equal content of polygons
- $\S 19$. Parallelograms and triangles having equal bases and equal altitudes
- $\S 20$. The measure of area of triangles and polygons
- $\S 21$. Equality of content and the measure of area
- CHAPTER $\text V$. DESARGUES'S THEOREM.
- $\S 22$. Desargues's theorem and its demonstration for plane geometry by aid of the axioms of congruence
- $\S 23$. The impossibility of demonstrating Desargues's theorem for the plane without the help of the axioms of congruence
- $\S 24$. Introduction of an algebra of segments based upon Desargues's theorem and independent of the axioms of congruence
- $\S 25$. The commutative and the associative law of addition for our new algebra of segments
- $\S 26$. The associative law of multiplication and the two distributive laws for the new algebra of segments
- $\S 27$. Equation of the straight line, based upon the new algebra of segments
- $\S 28$. The totality of segments, regarded as a complex number system
- $\S 29$. Construction of a geometry of space by aid of a desarguesian number system
- $\S 30$. Significance of Desargues's theorem
- CHAPTER $\text {VI}$. PASCAL'S THEOREM.
- $\S 31$. Two theorems concerning the possibility of proving Pascal's theorem
- $\S 32$. The commutative law of multiplication for an archimedean number system
- $\S 33$. The commutative law of multiplication for a non-archimedean number system
- $\S 34$. Proof of the two propositions concerning Pascal's theorem. Non-pascalian geometry
- $\S 35$. The demonstration, by means of the theorems of Pascal and Desargues, of any theorem relating to points of intersection
- CHAPTER $\text {VII}$. GEOMETRICAL CONSTRUCTIONS BASED UPON THE AXIOMS $\text I$–$\text V$.
- $\S 36$. Geometrical constructions by means of a straight-edge and a transferer of segments
- $\S 37$. Analytical representation of the co-ordinates of points which can be so constructed
- $\S 38$. The representation of algebraic numbers and of integral rational functions as sums of squares
- $\S 39$. Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments
- Conclusion