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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 Göttingen 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 Göttingen, 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. Among the important results obtained, the following are worthy of special mention:
This development and discussion of the foundation principles of geometry is not
only of mathematical but of pedagogical importance. Hoping that through an English
edition these important results of Professor Hilbert’s investigation may be made more
accessible to English speaking students and teachers of geometry, I have
undertaken, with his permission, this translation. In its preparation, I have
had the assistance of many valuable suggestions from Professor Osgood of
Harvard, Professor Moore of Chicago, and Professor Halsted of Texas. I am also
under obligations to Mr. Henry Coar and Mr. Arthur Bell for reading the
proof.
E. J. Townsend
University of Illinois.
“All human knowledge begins with intuitions, thence passes to concepts and ends with ideas.”
Kant, Kritik der reinen Vernunft, Elementariehre, Part 2, Sec. 2.
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