# Does Gauss own two “Theorema”?

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?

• Gauss called it egregium; see Disquisitiones generales circa superficies curvas (1827), page 24. – Mauro ALLEGRANZA Jan 2 '17 at 11:40
• What is your source for "elegantissimum" ? – Mauro ALLEGRANZA Jan 2 '17 at 11:41
• de.wikipedia.org/wiki/… in german. It is different from egregium, it is relate to gauss-bonnet. – Upc Jan 2 '17 at 11:45
• 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 ? – Mauro ALLEGRANZA Jan 2 '17 at 11:52
• – Mauro ALLEGRANZA Jan 2 '17 at 12:20

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).

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