General Investigations of Curved Surfaces of 1827 and 1825
General Investigations of Curved Surfaces of 1827 and 1825
复制标题
1827 年和 1825 年曲面的一般研究
作者：
C. Gauss
这是高斯关于曲面理论的经典回忆录的英译本。第一篇论文是1827年提交给皇家哥廷根学会的，至今仍被认为是研究无穷小几何最完整和最有用的导论。译文是以原文的复印件为基础的，但在编写原文和本说明的工作中参考了所有其他版本。随后是由高斯提交给哥廷根皇家学会的摘要的翻译。在题为《新的曲面研究》的标题下，译者接下来给出了一篇真正由高斯在更早的日期(1825年)写的论文，但直到1900年高斯作品的第八卷出现时，这篇论文才出版。这两篇论文都包含了现在所知的高斯曲率度量的基本性质，测地多边形的球面过剩与辅助球面上的相应面积成比例的定理，以及从固定点到测地线距离相等的点的轨迹与这些测地线垂直相切的证明。然而，1825年的论文包含了被高斯在1827年的论文中省略的关于平面曲率的介绍性材料，尽管在1825年，高斯只使用了测地极坐标，但在1827年，他引入了广义坐标p和q的概念。因此，可以看出，论文的印刷顺序与时间顺序相反。附注和参考书目：James Caddall Morehead，M.S.和Adam Miller Hiltebeitel，A.P.Viii+127。(美国新泽西州普林斯顿：普林斯顿图书馆出版协会，1902年。)价格1.75美元。
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.