Calculation of Gauss leading to 18:7 resonance between orbits of Jupiter and Pallas

Question

After Gauss helped relocate Ceres, he studied the orbit of the asteroid Pallas and discovered (1812) that Jupiter and Pallas have an orbital resonance that is nearly equal to 18:7. For instance, using the modern estimates of their orbital periods as 4332.59 days and 1684.87 days respectively, their ratio has continued fraction expansion $[2,1,1,2,1,511,2,\ldots]$, which is quite clearly approximated well by $[2,1,1,2,1] = 18/7$. Of course Gauss did not have the modern values for the orbital periods. If we use the cruder estimates of 4333 and 1685 we get a ratio with continued fraction expansion $[2, 1, 1, 2, 1, 240, 113, \ldots]$, which again is begging to be approximated by $[2,1,1,2,1]$.

My questions:

1. What was the ratio computed by Gauss that he then determined is nearly 18/7? Maybe it was not the orbital period directly but some other astronomical measure whose ratio would come out to the same thing.
2. Did he use continued fractions, or was his great familiarity with decimal expansions of "small" fractions enough? I think this example is a really super illustration of continued fractions, but I would like to know for sure if it's how Gauss attacked the problem.

I can find a number of sources discussing this resonance, but they mention the value 18/7 without giving the Gaussian calculation behind it (i.e., what ratio did he estimate as 18/7), and they don't indicate his method either.

Answer 1

Gauss's ratio was that of the mean motions $\mathit{[M]}$ (⚴) and $\mathit{[M']}$(♃) of Pallas and Jupiter. (Instead of these letters he used planet symbols, which are shown in their unicode version in brackets.)

As can be seen in his Nachlass (Werke, vol. 7, e.g. p. 553), he was expressing the perturbations of Pallas elements as sums of dozens of trigonometric terms $A\sin(k\mathit{[M']-\ell [M]}+\delta)$. On p. 604 the editors comment:

On one of the sheets which contain the integration of the perturbations of the epoch $\varepsilon$, one finds the following small computation, in which the first number given is Pallas's mean motion $\mathit{[M]}$: $$ \begin{gather} \mathit{769'',202079}\\ \textit{das 7-fache} = \mathit{5384'',414553}\\ \underline{\mathit{18\ m. m. [M'] = 5384'',392272}}\\ \mathit{18 [M']- 7[M] = -0,022281} \end{gather} $$ This seems to be the only hint at the first step in Gauss's discovery of the commensurability of both periods; from this one can't however conclude anything more than that he had just found the quantity $\mathit{18 [M']- 7[M]}$ extremely small.

I read this as saying that this ("small divisor"?) stood out enough as it is, numerically among the $\mathit{k[M']- \ell[M]}$'s he was looking at, with no need for an appeal to, e.g., continued fractions.