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This answer to the question First observation that the movement of a planet or asteroid in its orbit was affected by another planet says:

In 1705, with the mathematical assistance of Issac Newton, Edmond Halley published Synopsis of the Astronomy of Comets which made the prediction that the comet observed in 1682 had previously been observed in 1531 and 1607 was the same object returning about every 76 years or so. He included a rough estimation of the gravitational effects of Jupiter and Saturn, and predicted it would return in 1758.

Is there anything written specifically about mathematics used in a "rough estimation of the gravitational effects of Jupiter and Saturn"? Was the size of the effect on the comet's orbit mentioned, or anything about how the calculation was made?

Just fyi the linked answer also cites Wikipedia's Halley's Comet and the passage cites page 86 of P. Lancaster-Brown (1985). Halley & His Comet. Blandford Press. ISBN 0-7137-1447-6.

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    $\begingroup$ Strong suspicion here that the reply attributing to Halley 'a rough estimation of the gravitational effects of Jupiter and Saturn' makes a confusion with the 1750s work of Clairaut et al. Halley's contributions seem to have been to identify the probable periodic nature of a number of comets, and to calculate them on a parabolic or possibly elliptical basis. I can't find any early source showing he carried out calculations of planetary perturbations, the (recent) Lancaster-Brown book that has been cited in wikipedia for this is not available to me and I can't identify what L-B's source was. $\endgroup$
    – terry-s
    Mar 11, 2019 at 7:56
  • $\begingroup$ @terry-s my primary interest is if anyone did a calculation of this type as part of a prediction of the return time. I'll modify the question to open it up. Thanks! $\endgroup$
    – uhoh
    Mar 11, 2019 at 8:00

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Alexis Clairaut designed and led the detailed calculations of planetary perturbations to estimate the return-date in 1759 of what then became known as Halley's comet. The most detailed account of the work (in which Clairaut was assisted in the calculations by the young Jerome de Lalande and by Mme Lepaute) seems to be that given by Curtis Wilson in the Journal for the History of Astronomy v.24 (1993) pp.1-15: "Clairaut's Calculation of the Eighteenth-century Return of Halley's Comet".

The full text is available online. (I can hardly add anything except to mention that while Wilson cites early sources, he does not attribute to Halley any calculation of the planetary perturbations of the comet by Jupiter or Saturn. It seems that Halley's contribution here was indeed to identify the probable periodic nature of a number of comets, and to calculate their orbital paths on an unperturbed parabolic and then elliptical basis.)

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  • $\begingroup$ Oh this is fascinating reading, thank you! Even the first sentence of the abstract makes it clearer what the calculation might have been like: The calculation by Alexis-Claude Clairaut (1713-65), the French mathematician and sometimes prodigy, of a perihelion date for the return of Halley’s Comet in the late 1750s involved the first large-scale numerical integrations ever performed. $\endgroup$
    – uhoh
    Mar 11, 2019 at 8:52
  • $\begingroup$ I think this answer would also be an excellent addition to those already to Remarkable numerical calculations before electronic computers update: oh, you've got it there already, nicely done! $\endgroup$
    – uhoh
    Mar 31, 2019 at 3:48

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