To get an intuitive grasp of Gauss' notion of "curvature," try wrapping a piece of paper around a ball. The paper crumples and does not wrap smoothly. As Gauss saw it, the intrinsic curvature of the piece of paper is different from that of the sphere.
A flat plane however will wrap smoothly on the circumference of a cylinder. The intrinsic curvature of the cylinder can therefore be said to be "zero," meaning it is equal to that of the plane.
Gauss measured changes in the intrinsic curvature of a surface by examining the length of infinitesimal line elements and how they changed from one point on the surface to another.
He realized that the relationship described by the Pythagorean theorem is true only on a flat surface. He could therefore measure changes in the intrinsic curvature of a line element by changes in the Pythagorean relationship.
If the diagonal length between two segments of longitude and latitude on a surface is extended by "ds," the measures of longitude and latitude will expand by corresponding small amounts "dx" and "dy," depending on the curvature.
(The Pythagorean relationship describing the length of ds will be ).
Gauss constructed a "parametric formula" describing how the differential elements changed with respect to each other on different parts of the surface.
On a flat plane Gauss' expression for the length of the line element reduced to the familiar Pythagorean formula, .
While Gauss limited his investigation of curvature to two dimensions, his student Bernhard Riemann generalized the idea to three (and higher) dimensions in his inaugural lecture at Göttingen University, On the Hypotheses which Underlie Geometry. A separate web page sketching out Riemann's ideas on hypergeometries is here.
In his General Theory of Relativity, Albert Einstein described the effect of gravitation as the equivalent of inertial motion along least-action pathways (geodesics) in a curved "Riemannian" spacetime manifold. Physical "forces" are a consequence of geometry in Einstein's theory.
For those interested in learning more about Gauss' theory, we recommend Peter Pesic's introduction to the new Dover edition of General Investigations of Curved Surfaces.
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