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.



Equivalent definitions of Gaussian curvature
I like the presentation of the Theorema Egregium in *A Comprehensive Introduction to Differential Geometry* (Volume 2) by Michael Spivak. A translation of the original paper by Gauss and the historical background can be found in *General investigations of curved surfaces of 1827 and 1825* by James Caddall Morehead and Adam Miller Hiltebeitel (the 2005 Dover edition). 


