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Historically, Gauss measured it by an old method called after his name today: The Gauss map.

This maps the normal vector at each point of a surface curve to the unit sphere. Then he measured the angle difference between the starting normal at a point and a normal to the surface as it traced the sides of an infinitesimal triangle on the surface and mapped the corresponding normals on the unit sphere to find another triangle there. Finally, divided the corresponding triangle area on the unit sphere by the triangle area on the surface to find the curvature. This method appeared in his 1827 work "Disquisitiones generales circa superficies curvas" (General Investigations of Curved Surfaces)

Nevertheless, the same map was used by Olinde Rodrigues 12 years earlier than Gauss, anticipating him as early as 1815 by expressing the "Gaussian" curvature as the quotient of the aforementioned surfaces and proving that this curvature is equal to the product of principal curvatures.

It is not known today if Gauss was aware of Rodrigues work or discovered it independently but certainly, Olinde Rodrigues has not received the recognition that he deserves, and his work has not been sufficiently disseminated.

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    $\begingroup$ If by "discovering" you mean defining something equal to the section curvature extrinsically then Euler did that already back in 1760 when he multiplied the principal curvatures. Gauss is credited the most for discovering intrinsic geometry described by the first fundamental form, and proving that the curvature only depends on the latter (theorema egregium) despite the appearances in constructions that employ 3D space, see e.g. Struik, Outline of a History of Differential Geometry (II). But I agree that Rodrigues's work should be better known. $\endgroup$
    – Conifold
    Apr 22 at 3:53
  • $\begingroup$ No, I mean exactly what the question asks. The first and second fundamental forms are an historic serendipity and aporrhoea of the calculations to measure "Gaussian" curvature at a point by using Rodrigues' map. $\endgroup$
    – dimachaerus
    Apr 22 at 18:58
  • $\begingroup$ Rodrigues should be mentioned between Euler and Gauss since his construction gave a direct geometric interpretation to Euler's algebraic coefficient. Unfortunately, this is rarely done. But his construction does not establish that section curvature is intrinsic, which is what is usually meant by "discovering" it. So the bulk of the "discovery" credit still goes to Gauss. $\endgroup$
    – Conifold
    Apr 22 at 20:59
  • $\begingroup$ Do you have any idea where I can find Rodrigues' memoir? It must have been written circa 1815-1817. Is it: "Memoire sur les surfaces courbes" (Memoir on curved surfaces) which was published in 1817? $\endgroup$
    – dimachaerus
    Apr 22 at 21:35
  • $\begingroup$ Rodrigues, O., 1815. Recherches sur la théorie analytique des lignes et des rayons de courbure des surfaces, et sur la transformation d’une class d’intégrales doubles, qui ont un rapport direct avec les formules de cette théorie. Correspondance sur l’École Polytechnique 3, 162–182. Some volumes of Correspondance have been digitized, but I did not find this one online. $\endgroup$
    – Conifold
    Apr 22 at 22:26

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