 Gauss worked with coordinate systems within the surface itself, and considered the distance between points on the surface S along lines contained entirely in S. Now, some surfaces \(S_1\), \(S_2\) can be obtained from each other by bending (without stretching); expressed more formally, there is an isometric map from \(S_1\) to \(S_2\), which means that the points on \(S_1\) are assigned points in \(S_2\) so that distances within the surface \(S_1\) are the same as the distances of the corresponding points in \(S_2\). (For instance, a plane can be so bent to form of cylinder; you may use a piece of paper to convince yourself.)  And Gauss made the remarkable discovery that the Gaussian curvature of a surface is an intrinsic property that remains invariant under isometric mappings. He was so happy about this result that he called it theorema eggregium, his ‘outstanding theorem’: if two surfaces \(S_1\), \(S_2\) are isometric, then corresponding points have the same Gaussian curvature. 