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I am looking for the historical context in which Gauss established its famous Theorema Egregium. Was Gauss studying map projections (a nowadays popular application of the theorem)? Any references are welcome!

Mauricio
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Humberto José Bortolossi
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  • C. F. Gauss, "Disquisitiones generales circa superficies curvas." Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores, Vol. 6 for the years 1823-1827. Göttingen: Dieterich 1828, pp. 99-146 (presented to the society on October 8, 1827). On p. 120: " THEOREMA. Si superficies curva in quamcunque aliam superficiem explicatur, mensura curvaturae in singulis punctis invariata manet. " – njuffa Aug 19 '23 at 06:18
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    This is likely worth reading: Peter Dombrowski, "150 years after Gauss' « disquisitiones generales circa superficies curvas »". Astérisque, no. 62 (1979), 157 pp. (online). There is a section: On the history of the origin of the "Disquisitiones Generales" and on the history of the Theorema Egregium – njuffa Aug 19 '23 at 10:55

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The background, and the applications, have to do with practical surveying of the earth's surface. One of the uses Gauss makes of the theorema egregium is a comparison theorem for small triangles when one compares a triangle in the surface with the corresponding triangle of the same sidelengths in the plane. This is discussed in Dombrowski, pages 114-115. The earth is less curved toward the poles, with the result that the angular correction will be smaller for vertices closer to the poles:

Gauss gives these different correction values according to (35) for one of the largest terrestial triangles measured by him, namely with "vertices" at Brocken, Hohehagen and Inselsberg (Dombrowski p. 115).

Mikhail Katz
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  • In fact considerations about the area of curved triangles on surfaces is what lead Gauss to the first proof about Theorema Egregium. Such proof, however, relied on a subtle limit procedure which, in the shaky foundation of calculus of the first half of XIX century,Gauss didn't feel safely sure about. It is notorious he always refrained from making statements about which he was not 100% sure. That's why he developed his proof relying on a long and ingenious computations, which he probably would have not pursued if he wasn't positive about its outcome. – Nicola Ciccoli Sep 13 '23 at 08:53
  • "shaky foundations ... Gauss didn't feel safely sure about": apart from the fact that this sort of claim is "common knowledge", do you have any evidence for the claim that Gauss thought foundations were "shaky" and that he "didn't feel safely sure"? Otherwise such claims risk being triumvirate history. @NicolaCiccoli – Mikhail Katz Sep 13 '23 at 10:24
  • We know that Gauss knew the invariance of curvature under isometries already around 1816-18 (Stackel, Dombrowski) since he was able to derive it from the computation of angular defects of curved triangles: call this the "geometric proof". However he choose to publish it only when he was able to give an analytic formula, the one he gave in Disquisitiones. The "geometric proof" has to rely on the fact that the curvature at a point seen as a limit of ratio of areas is independent of the way in which this limit is done. – Nicola Ciccoli Sep 13 '23 at 13:19
  • (as in Spivak: "it is not a priori clear whether this limit exists, or if it depends on the way in which the area approaches the point") and that any curve on the surface can be approximated by a polygonal. If I remember correctly both Dombrowski and this paper by Cogliati speculate that he preferred a purely analytical proof since it freed the geometric one from this limit procedures (which are indeed non trivial even in modern terms). That's all I meant. No knowledge about trimuvirate history whatever it means. Cannot be more precise since I do not have original works with me. – Nicola Ciccoli Sep 13 '23 at 13:19
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    A. Cogliati, Sulla ricezione del Theorema Egregium, Storia delle Scienze Matematiche 2018. – Nicola Ciccoli Sep 13 '23 at 13:20
  • @NicolaCiccoli, Thanks for the reference, but my Italian is very weak and I have to rely on my coauthors when I need a passage translated. What evidence does Cogliati present there? – Mikhail Katz Sep 13 '23 at 13:25
  • @NicolaCiccoli, I read some of Dombrowski's article though not all. Does he assert that Gauss was reluctant to rely on the geometric proof because of a concern with the validity of the limit process? As you mention above, Cogliati only "speculates" :-) Spivak has an interest in history but he is not a historian, so whatever he claims should probably be taken with a grain of salt. – Mikhail Katz Sep 13 '23 at 13:27