I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, that is the Gaussian curvature of a surface is an intrinsic quantity. 


For instance, I am fascinated by whether Gauss had imagined that it was an intrinsic property or, after a lengthy calculation, he found out it was. Perhaps the fact that he called this result Remarkable Theorem points toward the latter.


I have not been able to find a book on the history of differential geometry that would adress this. More generally, I would like to know more about the history of differential geometry and I would welcome any suggestions for books or surveys on it.


Thanks



History surrounding Gauss Theorema Egregium and differential geometry
It seems to me that Volume 2 of Mike Spivak course on differential geometry gives an answer to your question. See Section A of chapter 3 : How to read Gauss?


