I get the below from Neptune and Pluto.

In 1821, Alexis Bouvard published astronomical tables of the orbit of Neptune's neighbour Uranus.[22] Subsequent observations revealed substantial deviations from the tables, leading Bouvard to hypothesise that an unknown body was perturbing the orbit through gravitational interaction.[23] In 1843, John Couch Adams began work on the orbit of Uranus using the data he had. Via Cambridge Observatory director James Challis, he requested extra data from Sir George Airy, the Astronomer Royal, who supplied it in February 1844. Adams continued to work in 1845–46 and produced several different estimates of a new planet.

In the 1840s, Urbain Le Verrier used Newtonian mechanics to predict the position of the then-undiscovered planet Neptune after analysing perturbations in the orbit of Uranus.[26] Subsequent observations of Neptune in the late 19th century led astronomers to speculate that Uranus's orbit was being disturbed by another planet besides Neptune.

Does anyone know how did people calculate their orbits in 19th century? It is not a three-body problem, right?

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    $\begingroup$ Can't say for certain, by my assumption is that they defined the orbit in terms of the common orbital elements and used the standard equation for elliptical motion, then added perturbations to that equation based on the known perturbation effects from the other planets. Once you have that (complicated) equation, you try to fit it to your data. If you can't, you say, what additional perturbation can I add that will fit the data and then assume that is due to an unknown planet. $\endgroup$
    – zephyr
    Dec 15 '16 at 13:52
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    $\begingroup$ "Pluto was found, its faintness and lack of a resolvable disc cast doubt on the idea that it was Lowell's Planet X", but, to semi-answer your question, they probably solved the differential equations numerically, which is painful without computers, but still possible. $\endgroup$
    – user21
    Dec 15 '16 at 16:19
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    $\begingroup$ I came across Le Verrier's work here. Its in French, but the mathematical formulas can be followed. play.google.com/books/… I don't think 3 body mathematics was necessary, but regression analysis where Uranus speeds up as it approaches Neptune and slows down after it passes. Wikipedia mentions regression analysis en.wikipedia.org/wiki/… $\endgroup$
    – userLTK
    Dec 16 '16 at 12:26
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    $\begingroup$ Found Adams work, if interested: archive.org/stream/explanationobse00Adam#page/13/mode/1up Props to you if you have the patience to work though their calculations. $\endgroup$
    – userLTK
    Dec 16 '16 at 12:42
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    $\begingroup$ @questionhang looking at NASA's JPL site: ssd.jpl.nasa.gov/horizons.cgi#top It looks like Uranus passed Neptune around 1993 and Uranus catches up to Neptune every 171 years, which would make the previous pass around 1822, so in the 1840s, Neptune should have been slowing down Uranus, especially if some of the data they looked at was from the 1820s, which I think it was. Also, They are not considered to be in 2:1 resonance. They are close, but the 6 year variation per pass is consistent. Resonance requires a back and forth, moving ahead then falling behind the resonance. $\endgroup$
    – userLTK
    Dec 17 '16 at 4:37

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