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The material contained in the following translation was given in substance by Professor Hilbert as a course of lectures on euclidean geometry at the University of Gottingen during the winter semester of 1898–1899. The results of his investigation were re-arranged and put into the form in which they appear here as a memorial address published in connection with the celebration at the unveiling of the Gauss-Weber monument at Gottingen, in June, 1899. In the French edition, which appeared soon after, Professor Hilbert made some additions, particularly in the concluding remarks, where he gave an account of the results of a recent investigation made by Dr. Dehn. These additions have been incorporated in the following translation. 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.
1. The Five Groups of Axioms,
2. Compatibility and Mutual Independence of the Axioms,
3. The Theory of Proportion,
4. The Theory of Plane Areas,
5. Desargues’s Theorem,
6. Pascal’s Theorem,
7. Geometrical Constructions Based Upon the Axioms I–V.