When I read our differential geometry book, I saw two theorema: "Theorema Egregium" and "Theorema Elegantissimum". Mathematically, they are not the same. On wikipedia, there is nothing about Elegantissmum. What is the story behind it?
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$\begingroup$ Gauss called it egregium; see Disquisitiones generales circa superficies curvas (1827), page 24. $\endgroup$– Mauro ALLEGRANZACommented Jan 2, 2017 at 11:40
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1$\begingroup$ What is your source for "elegantissimum" ? $\endgroup$– Mauro ALLEGRANZACommented Jan 2, 2017 at 11:41
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$\begingroup$ de.wikipedia.org/wiki/… in german. It is different from egregium, it is relate to gauss-bonnet. $\endgroup$– UpcCommented Jan 2, 2017 at 11:45
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$\begingroup$ Thanks, I've seen it... bbut I'm not able to read German and on other Wiki versions : English, French, Italian there is no such title. Are there in German paragraph more info available ? $\endgroup$– Mauro ALLEGRANZACommented Jan 2, 2017 at 11:52
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$\begingroup$ Related post : did-gauss-formulate-or-at-least-know-of-the-full-essence-of-the-Gauss-Bonnet-Theorem ?t. $\endgroup$– Mauro ALLEGRANZACommented Jan 2, 2017 at 12:20
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In his Disquisitiones generales circa superficies curvas (1827), §12, page 24, Gauss called egregium [sponte perducit ad egregium, i.e. spontaneously leads to excellent] the following Theorem:
Si superficies curva in quamcumque aliam superficiem explicatur, mensura curvaturae in singulis punctis invariata manet. [If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.]
And see §20, page 36 for a result regarding the theory of curved surfaces called elegantissima :
Excessus summae angolorum trianguli...
According to: John McCleary, Geometry from a Differentiable Viewpoint (2nd ed, 2013), the Gauss-Bonnet Theorem is linked to Gauss' "Theorema Elegantissimum", referring to Gauss' Disquisitiones, (1825, §20).
See also into Theoria residuorum biquadraticorum (1825), page 30, he calls elegantissimum the result:
$P \equiv 2a (\text {mod} p)$.
answered Jan 2, 2017 at 12:05