General Investigations of Curved Surfaces of 1827 and 1825

Abstract

THIS is an English translation of the classic memoirs of Gauss on the theory of surfaces. The first paper is that which was presented to the Royal Society of Göttingen in 1827, and is still regarded as the most finished and useful introduction to the study of infinitesimal geometry. The translation is based on a copy of the original paper, but in the work of preparing it and the present notes all the other editions were consulted. This is followed by a translation of the abstract presented by Gauss to the Royal Society of Göttingen. Under the title of “New General Investigations of Curved Surfaces,” the translators next give a paper really written by Gauss at an earlier date (1825), but which was not published until the eighth volume of Gauss's works appeared in 1900. Both papers contain the fundamental properties of what is now known as Gauss's measure of curvature, the theorem that the spherical excess of a geodesic polygon is proportional to the corresponding area on the auxiliary sphere, and the proof that the locus of points the geodesic distances of which from a fixed point are equal cuts these geodesies orthogonally. The paper of 1825, however, contains introductory matter on curvature in a plane which was omitted by Gauss from his 1827 paper, and while, in 1825, Gauss used geodesic polar coordinates only, in 1827 he introduced the notion of generalised coordinates p and q. It will thus be seen that the order in which the papers have been printed is the reverse of chronological order.

General Investigations of Curved Surfaces of 1827 and 1825.

By Karl Friedrich Gauss. Translated with Notes and a Bibliography by James Caddall Morehead, A.M., M.S., and Adam Miller Hiltebeitel, A.M. Pp. viii + 127. (Princeton, N.J., U.S.A.: The Princeton Library Publishing Association, 1902.) Price 1.75 dollars.