(This is just to supplement the existing answers.)
A couple of additional useful references:
George E. Smith, Closing the Loop: Testing Newtonian Gravity, Then and Now, appearing as Ch. 10 in "Newton and Empiricism" (2014) by Zvi Biener and Eric Schliesser.
On p. 282 there's a neat plot showing the approx. 900-yr Jupiter-Saturn "Great Inequality" over the past 2000 yrs. (Fluctuation in Saturn's motion peaks at about 60 minutes of arc and Jupiter's peaks at about 20 minutes of arc.) Here's the gist of Smith's explanation, in my own words:
A unique line L (a "diameter") can be drawn in the plane of the orbits of Jupiter and Saturn, passing through their points of least- and greatest- separation. Conjunctions are then of two types: (a) for conjunctions that occur on one side of line L, the planets have greater separation after the conjunction than before, and (b) for conjunctions on the other side of line L, the planets have less separation after the conjunction than before. Conjunctions occur about once every 20 years, with Saturn covering about 2/3 of its orbit in that time; consequently, about 2 of every 3 conjunctions occur on the same side of line L, producing a net perturbation of both planets in the course of every 3 conjunctions -- about every 60 years. It so happens that about 450 years of conjunctions are required before the "2 of every 3" switch types (i.e. change the side of line L on which they occur), thus reversing the effect and resulting in a cycle whose period is about 900 years.
And pp. 298ff. discuss the "secular" acceleration of the moon's mean motion. The gist here is that the moon's mean motion is observed to be accelerating about 12 arc-seconds/century, an amount explained by two theoretical components:
- About 6 arc-seconds/century due to planetary perturbations of Earth's orbit. (This is a sum of perturbation terms whose dominant periodic part was found by Laplace to be about 10 arc-seconds/century; Adams later included higher-order terms whose effect was to reduce the total to the currently accepted 6 arc-seconds/century.)
- About 6 arc-seconds/century due to tidal effects via the moon's gravity.
For the period as found by Laplace, I've been unable to find a source that provides anything more precise than just "millions of years". I had hoped to find the value in Laplace's actual derivation at the Smithsonian Library, but alas, my French is lacking.
Kushner, David. The Controversy Surrounding the Secular Acceleration of the Moon’s Mean Motion, Archive for History of Exact Sciences, vol. 39, no. 4, Springer, 1989, pp. 291–316.
I didn't know this when I asked, but apparently the second part of my question was at one time a very hot potato:
But one of these secular inequalities has particularly engaged the attention--and enraged the passion--of astronomers: the secular variation of the moon's mean motion. Indeed the international controversy which flared up circa 1860 was one of the largest and most active of the century.
Finally, a quote that I enjoyed (from Smith, p. 299):
"For I find this Theory [of the Moon] so very intricate & the Theory of Gravity so necessary to it, that I am satisfied it will never be perfected but by somebody who understands the Theory of gravity as well or better than I do."
-- Isaac Newton (in a letter to Flamstead, February 16, 1695)