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 $[2,1,1,2,1]$.
- 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.
- 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.