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I know that a special case of the Bonnet theorem, called the Theorema Elegantissimum, was proved by Gauss in his 1827 treatise on differential geometry. This was a theroem that dealt with the connection between total curvature and angular deficit, but only in the case of geodesic lines. I know also that he described the notion of geodesic curvatue in an unpublished paper. But is there a place (letters, private notes) where he directly referred to the full version of Bonnet theorem, e.g the connection between geometry and topology (in the sense of Euler's characteristic)?

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Apparently not. On page 463 of Daniel Gottlieb's 1996 article All the Way with Gauss-Bonnet and the Sociology of Mathematics (The American Mathematical Monthly, Vol. 103, No. 6), he writes (according to Morris Hirsch) that Walter Dyck in 1888 was the first to connect the degree of the Gauss map with the Euler-Poincaré characteristic. A few lines later in the same article, Gottlieb writes that Hans Samelson was unable to find a statement of the global Gauss-Bonnet theorem in Gauss' works.

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