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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

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3 Answers 3

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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?

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Thanks. As you said, the answer to my question is exactly on page 143 of Volume 2 of Mike Spivak course on differential geometry. According to Spivak, Gauss showed that the Gaussian curvature is intrinsic by taking the limit using the now called Gauss-Bonnet Theorem for geodesic triangles. Once he realized that, he then set out to do the computation in Theorema Egregium and of course he succeeded in carrying it out.

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You should read Gauss' book (General theory of curves and surfaces). It is written in a fairly unenlightening manner (back then, people did not trust pictures, so Gauss just does pages and pages of horrible computations), but if you read it carefully, you will see that he had a very geometric view of things. Secondary sources (e.g., Spivak) are based entirely on Gauss' book and since Gauss did not write an autobiography (to the best of my knowledge) you should go to the said book of his.

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3  
People trust pictures now? If the horrible computations are lessened now, perhaps it's just that we have better packaging. –  Ryan Reich Oct 11 '12 at 16:50
2  
It's not a question of trusting proof by picture (which, as far as I know, people still do not), but supplying pictures to explain what it is he is computing. –  Igor Rivin Oct 11 '12 at 17:00

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