The Foundations of Geometry
David Hilbert, trans. E.J.Townsend
From the introduction:
“As a basis for the analysis of our intuition of space, Professor Hilbert commences his discussion by considering three systems of things which he calls points, straight lines, and planes, and
sets up a system of axioms connecting these elements in their mutual relations. The purpose
of his investigations is to discuss systematically the relations of these axioms to one another
and also the bearing of each upon the logical development of euclidean geometry. Among the
important results obtained, the following are worthy of special mention:
1. The mutual independence and also the compatibility of the given system of axioms is
fully discussed by the aid of various new systems of geometry which are introduced.
2. The most important propositions of euclidean geometry are demonstrated in such a
manner as to show precisely what axioms underlie and make possible the demonstration.
3. The axioms of congruence are introduced and made the basis of the definition of geometric
4. The significance of several of the most important axioms and theorems in the development of the euclidean geometry is clearly shown; for example, it is shown that the whole of the
euclidean geometry may be developed without the use of the axiom of continuity; the significance of Desargues’s theorem, as a condition that a given plane geometry may be regarded as a
part of a geometry of space, is made apparent, etc.
5. A variety of algebras of segments are introduced in accordance with the laws of arithmetic.”