 Some of the epistemological issues are already raised with Gauss’s work (1828), so we turn to it first. Euler had shown long before that, at each point P in a (smooth) surface, there are two so-called principal curvatures, the maximum and the minimum curvature of lines passing through P. These curvatures, \(\kappa_{1}\) and \(\kappa_{2}\), occur in plane sections perpendicular to each other and allow us to study the shape of the entire neighborhood of P. Now, Gauss’s deep idea (which remarkably he found while working on geodesy, i.e., surveying and mapmaking) was that some properties of curved surfaces can be studied without regard for the ambient space. And one of these key properties is the product \(\kappa_{1}\kappa_{2}\), called the Gaussian curvature of the surface (at a point). Thus, employing methods of differential analysis, he elaborated what is called an intrinsic geometry of surfaces. 