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In 1821 Alexis Bouvard published a book with tables of the orbits of Jupiter, Saturn and Uranus and future predictions of the orbits. The real orbit of Uranus deviated from the calculations which was the starting point of an investigation which eventually led to the discovery of Neptune.

I am wondering how he performed the calculations, which I presume are not just Kepler orbits, but including some pertubations from the known planets (correct me if I understand that wrong). Usually nowadays you would do pertubations calculations in the Hamilton-Jacobi framework, but since Jacobi just started college in 1821 it had not been invented yet (I presume).

So does anybody know how exactly these calculations were done at that time?

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Perturbation theory begins with Newton who tried to explain the motion of the Moon (this is the object in the solar system whose motion deviates most from Keplerian orbits). Newton essentially failed, and more advanced methods of perturbation theory were developed in 18th century, main contributions are due to d'Alembert, Clairaut, Euler and Mayer. This resulted in the tables of Moon motion which had accuracy sufficient for navigation (that is an error of fraction of a minute over several years). Same ideas were applied to the large planets, of which the most conspicuous case is the 3-body problem Sun-Jupiter-Saturn. The perturbation theory is too technical to be explained here (see the reference below), but the modern notions of Hamilton-Jacobi framework slowly evolved in the process of this development of perturbation theory. This is a general pattern: scientists solve concrete problems, and only after that their methods are generalized into some general framework.

Ref. Curtis Wilson, The Great Inequality of Jupiter and Saturn: from Kepler to Laplace, Arch. Hist. Exact Sci., 33, 1/3, 1985, 15-290. Contains a popular not very technical historical exposition of perturbation theory, as it developed in the investigation of Jupiter and Saturn.

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  • $\begingroup$ Thanks. You state that the pertubation theory is too technical to be explained here (the technical part is what interests me most) and see reference below. But for the reference below you say it is no lt technical. Could you explain? $\endgroup$
    – lalala
    Commented Apr 13, 2023 at 18:57
  • $\begingroup$ The reference is 275 pages long. Do you want me to explain it here?? For full explanation, read the paper of Bouvard. $\endgroup$
    – Alexandre Eremenko
    Commented Apr 14, 2023 at 5:10
  • $\begingroup$ Where's your evidence that d'Alembert et al reached accuracies of a 'fraction of a minute' over years? (As you may know, Mayer's theory was recognized as best of that bunch, & he only claimed better than about 2' (Mayer, 1753), while Delambre's history (publ.1827) regretted the others had not tried harder to represent the moon's motion.) And where's your evidence for Newton's 'failure'? He was recognized (e.g. Laplace, Tisserand) for taking first steps in perturbational methods (Princ. 1.66, 3.25-9), though his results were limited by suboptimal math tools & a pioneer's approximations. $\endgroup$
    – terry-s
    Commented Mar 6 at 2:28
  • $\begingroup$ @terry-s: Newton indeed made the first step (he gave an explanation of the first inequality for the Moon). His attempts wo explain the other two inequalities known at that time failed. Mayer claimed 2' accuracy because this was sufficient to qualify for the Longitude Award. And he was indeed given a part of this award. He developed usable tables, on the basis of theories of Clairault and d'Alembert. Euler also received a part of the prize. In fact, Mayer's tables gave accuracy better than 1', and they were used in practical navigation. $\endgroup$
    – Alexandre Eremenko
    Commented Mar 6 at 13:03
  • $\begingroup$ @Alexandre Eremenko, you clearly consulted poor-quality sources. Known before Newton's work were at least 5 inequalities (elliptical, evection, variation, annual equation, and the latitude inequalities). He also newly found some others. The citations I gave before, show that besides a math/phys account of the ellipse Newton gave rather full derivation of the variation and the latitude inequalities, besides briefer explanations of the others. Although his work on perturbations was subject to approximations, his results are still recognisably represented in modern lunar theories. $\endgroup$
    – terry-s
    Commented Mar 15 at 17:57

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