Does Gauss own two “Theorema”?

Question

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?

Answer 1

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).

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See also into Theoria residuorum biquadraticorum (1825), page 30, he calls elegantissimum the result:

$P \equiv 2a (\text {mod} p)$.