would tend to a definite limit
when the points were taken closer and closer together. In particular we assumed that for infinitesimal areas of the surface the Pythagorean theorem
would hold. This was equivalent to stating that infinitesimal areas of our curved surface could be regarded as flat or Euclidean, hence as identified with the plane lying tangent to them. Inasmuch as the restriction of Euclideanism in the infinitesimal is precisely one that Riemann has imposed on space, we need have no fear of applying the differential method to the types of non-Euclidean space thus far discussed.
Nevertheless, we may conceive of spaces where
would not tend to a limit, and where, however tiny the area of our surface, we should still be faced with waves within waves ad infinitum. The situation would be similar to that presented by curves with no tangents at any point. However, such cases are only of theoretical interest.
[29] It should be mentioned that there exists another type of curvature, different from the Gaussian curvature. This other type of curvature is called the mean curvature at a point and is given by
