Pythagoras, made local.

Along a curve, ds² = dx² + dy² is Pythagoras for an infinitesimal step, and length is the limit of summed chords. On a surface the same rule becomes ds² = E du² + 2F du dv + G dv², and from those three numbers alone Gauss recovered the curvature: computed here from outside and from inside, then totalled over closed surfaces and followed along geodesics.