  We embark in a program of studying the problem of better approximating
surfaces by triangulations(triangular meshes) by considering the approximating
triangulations as finite metric spaces and the target smooth surface as their
Haussdorff-Gromov limit. This allows us to define in a more natural way the
relevant elements, constants and invariants s.a. principal directions and
principal values, Gaussian and Mean curvature, etc. By a "natural way" we mean
an intrinsic, discrete, metric definitions as opposed to approximating or
paraphrasing the differentiable notions. In this way we hope to circumvent
computational errors and, indeed, conceptual ones, that are often inherent to
the classical, "numerical" approach. In this first study we consider the
problem of determining the Gaussian curvature of a polyhedral surface, by using
the {\em embedding curvature} in the sense of Wald (and Menger). We present two
modalities of employing these definitions for the computation of Gaussian
curvature.
