Yesterday I took my time to look again into Schlesinger's essay on Gauss's contributions to analysis, and I found something new I didn't know about (so it caught my eye) in the last subsection of the chapter "The posthumous treatise on the agM [apparently 'arithmetico-geometric Mean']. Extension to the theory of module function" (p. 84-117). In subsection (e) of this chapter, named "The last diary notes from 1800. Asymptotic. Midpoint equation. Class number", Schlesinger comments on the equation of the center, includes some intriguing results of Gauss, and mentions that his formulas agree with that of Jacobi (1849). The relevant pages in Gauss's nachlass are p. 420-428 of volume 10-1 of his works, and they are entitled "on the convergence of expansions [in series] of the equation of the center" (German: 'Über die Konvergenz der Entwicklung der Mittelpunktsgleichung'). It seems that these pages are actually part [VII] of a planned treatise on the convergence of infinite series.
Just to be clear, I know nothing about this so called "Mittelpunktsgleichung", and I'm not even sure if this work was done by Gauss in astronomical contexts or analytic contexts, or perhaps both (I mention astronomical context since it seems that these formulas were connected with his work on "Kepler's equation" in astronomy).
So I'm not looking for a very sophisticated answer, just an explanation of what Gauss actually did and the context it was done. I think that if Schlesinger commented on it then it's important.