German mathematician and scientist Gauss (1777- 1855) is called the three greatest mathematicians in history together with Newton and Archimedes. Gauss is one of the founders of modern mathematics, and he has a great influence in history. He can be juxtaposed with Archimedes, Newton and Euler, and is known as the "prince of mathematics".

1828, Gauss published "General Theory of Surfaces", which comprehensively and systematically expounded the differential geometry of spatial surfaces and put forward the theory of intrinsic surfaces. Gaussian surface theory was later developed by Riemann.

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Since 18 16, Gauss has devoted most of his energy to the study of geodesy and cartography. In this respect, he published many papers, which aroused his interest in differential geometry, and thus wrote a paper published in 1827: Discontinue s generales circa superficies curvas. In addition to this paper on differential geometry of surfaces in three-dimensional space, Gauss also introduced a brand-new concept, that is, treating surfaces themselves as the same space. It is this concept that, after Riemann's popularization, finally opens up a brand-new prospect for non-Euclidean geometry.

Gauss's definition of curvature is the continuation of exponential curvature. Gauss also proved that if two surfaces can correspond to each other one by one and the distance elements of the corresponding points on the two surfaces are equal, then we call them equidistant surfaces and they must have the same geometry. In particular, they must have equal total curvature at corresponding points. If we want to move one part of a surface (at a certain distance) to another part, a necessary condition is that the curvature of this surface is constant. Therefore, a part of a sphere (the curvature is the reciprocal of the radius square) can be moved to another part without distortion, but this is not feasible for an elliptical sphere (in any case, as long as the curved surface or a part thereof is properly placed and reflected to another part).

Another major topic that Gauss studied in the paper of 1827 is: finding geodesics on surfaces. He also proved a triangle theorem about curvature and geodesic, which can explain that the triangle integral value of curvature in a geodesic (line) is equal to the triangle (inside) angle and the remainder of 180 degrees, or the sum of angles is less than 180 degrees. In addition, Gauss discussed the analytic problem of conformal mapping from one surface to another in Differential Geometry, and won an award from the Royal Danish Scientific Society 1822. So we say that Gauss's originality in differential geometry is undoubtedly a milestone in differential geometry itself. What's more, Gauss's works include that when the surface itself is regarded as a space, there is indeed non-Euclidean geometry on the surface. We don't know if Gauss has foresight in this non-Euclidean interpretation of surface geometry.

http://www2.emath.pu.edu.tw/s9005 153/Gauss-s.htm