Book:David Hilbert/The Foundations of Geometry

David Hilbert: Foundations of Geometry

Published $\text {1902}$ (translated by E.J. Townsend)

Subject Matter

• Euclidean Geometry

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

Further Editions

• 1899: David Hilbert: Grundlagen der Geometrie