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Today if I want to calculate an elliptical orbit of some object due to the gravity of another, I use Runge-Kutta integration, and I can see the individual steps I use shown in that article.

Back "in the day..." Kepler, Newton, and others might have used what we now call Kepler's Equation: $M=E-e \sin(E)$ where given M one had to solve for E.

I would like to see how Kepler, Newton, and similar contemporaries actually performed the calculation of the position in the orbit for a specified time. Are there reproductions, or images of their notes, showing the actual steps each of them used to solve this equation?

Did Newton or Leibniz also use some form of integration as an independent method to obtain the motion of an elliptical orbit as well? Are there images or reproductions of those calculations available anywhere?

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    $\begingroup$ Newton didn't need to do what you seem to think he did. You're talking about the problem of finding the position for a given time, assuming Kepler's laws are true. Newton wasn't concerned with that. His big accomplishment was to prove Kepler's laws from his own laws of motion. $\endgroup$
    – user466
    Commented Mar 26, 2016 at 15:39
  • $\begingroup$ @BenCrowell - throughout his productive life, Newton had many different "big accomplishments", and quite a variety of concerns. He's one of the great "unpidgeonholeables" of all time. Considering his contributions to math, are you absolutely sure there is no evidence anywhere that he ever applied any of those techniques to the movement of planets or satellites? $\endgroup$
    – uhoh
    Commented Mar 27, 2016 at 1:42
  • $\begingroup$ @BenCrowell What you are trying to explain to me here is finally sinking in, one year later! I now understand what you mean "Newton didn't need to do what you seem to think he did." - see this. I should clean up my caveleir comment, but I'll leave it for a little while at least as a reminder to myself. Thanks again! $\endgroup$
    – uhoh
    Commented Mar 20, 2017 at 3:30
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    $\begingroup$ You may enjoy this article: How Gauss Became Famous: maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/… $\endgroup$
    – Escualo
    Commented Mar 20, 2017 at 3:31
  • $\begingroup$ @Escualo I can see that I'll enjoy that very much indeed - thanks! $\endgroup$
    – uhoh
    Commented Mar 20, 2017 at 3:32

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Goldstine, A History of Numerical Analysis from the 16th through the 19th Century (1977), describes Kepler's approach (p. 47), which may be found in Kepler's Epitome Astronomiae Copernicanae (1618), Ch. 4, Bk. V., pp. 665f. It is an iterative numerical algorithm Kepler called regula positionum. Goldstine describes the steps of an example, which begins on p. 686 of the Epitome, but does not develop the theory (probably because it was not seminal?). Goldstine's numbers do not always match Kepler's; he may have corrected them or taken them from an excerpt containing this section printed (and translated) in Great Books of the World, vol. 16 (1952), pp. 998-999 (Goldstine's reference).

Goldstine (p. 64) also refers to Newton's solutions, which may be found in Principia, Book I, Prop. XXXI (Prob. XXIII) and Scholium. (Note Goldstine's reference "Problem XXII" is a typo.)

Newton first gives a solution involving a rolling wheel, to which the elliptical orbit is fixed. The wheel is rolled a distance that depends on the time elapsed, while a point on the ellipse fixed to the wheel traces a cycloid. He then constructs the point on the orbit and shows it has swept out the correct area.

Then admitting that foregoing method is hard to solve, Newton gives a solution by approximation in the scholium. Newton's method is, what else, Newton's Method (often called the Newton-Raphson Method today), although it is presented in the geometric terms of the problem.

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  • $\begingroup$ Fantastic! This is fascinating. Thanks for finding these. The google link to Goldstine is helpful - and I've found a few copies in nearby libraries. Had I had the foresight to ask "*What method did Newton use to solve Kepler's equation?", you could have answered with two words! $\endgroup$
    – uhoh
    Commented Mar 27, 2016 at 18:41
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Kepler's Proofs will get you started on your quest.

This article mentions 987 folio pages of arithmetic; you should also look at the tables and methods of Copernicus.

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  • $\begingroup$ OK that is a useful starting "roadmap", thank you. I am wondering if those are really 987 pages of Kepler trying to figure out how to calculate the orbit of Mars. I'm only looking for a clear example of the actual calculation of a position, given a time (after periapsis) itself. I almost titled this question "How did Kepler actually solve Kepler's Equation" but then included Newton because I thought that it would improve the chances of a definitive answer. $\endgroup$
    – uhoh
    Commented Mar 25, 2016 at 23:18
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At the time of Kepler, not only Runge-Kutta, but the very notion of the differential equation was not avalailable:-)

If you really would like to see how Kepler calculated the orbits, why don't you look at his own work, Astronomia Nova, which is available in English translation?

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  • $\begingroup$ Thank you! Yep, I started with the link to the RK method in order to give a clear example of what I am after - an illustration of the actual method used to calculate position, given time. If the method Kepler used is known from Astronomia Nova or elsewhere, then I'm sure scholars of the history of science and math will have found it, written about it, and possibly captured examples in the form of images. That's what I need to find - as well as something analogous from Newton's solving of position, given time. $\endgroup$
    – uhoh
    Commented Mar 25, 2016 at 23:28

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