Curvature of surfaces in euclidean 3-space: philosophical analysis of Gauss' Theorema Egregium

This essay, an excerpt of the author's Ph.D. in Philosophy of mathematics (2012) thought of as being a companion to recent discoveries of new explicit Cartan geometry curvatures, analyzes how Gauss, after having devised the isometrically invariant character of curvature, struggled with elimination computations in order to transform the available extrinsic expression of curvature into a new beautifully general differential expression which depends only on second order derivatives of the coefficients E, F, G of the intrinsic metric. Systematic speculations about the essence of impersonal mathematical necessitation are steadily inserted.