I'm trying to find out more about geometry of surfaces and, in particular, Gaussian curvature. I understand that it can be defined in terms of the principal curvatures (extrinsically) and also intrinsically, and that the result that it can be equivalently defined in these two ways was a significant result. Does anyone know where I can find a nice, clear proof of the equivalence of these definitions and perhaps some historical/background information on their importance? Thanks in advance.