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It is amazing that Kepler determined his three laws by looking at data, without a calculator and using only pen and paper. It is conceivable how he proved his laws described the data after he had already conjectured them, but what I do not understand is how he guessed them in the first place.

I will focus in particular on Kepler's third law, which states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of the orbit.

I assume that Kepler was working with data about the planets only, plus our own moon, and the sun. I make this assumption because I don't think Kepler had data about other moons, comets, or asteroids, which had not been observed by telescope yet. If this is true, knowing that Neptune, Uranus, and Pluto were not yet discovered when Kepler was alive, this means Kepler had less than 9 data points to work with.

My friend claims that it is totally concievably how Kepler guessed this relationship (although he provides no method of how Kepler might have done it), and also that Kepler's observations are "not that hard". As a challenge, I gave my friend a data table with one column labeled $x$, the other $y$, and 9 coordinates $(x,y)$ which fit the relationship $x^4=y^3$. I said "please find the relationship between $x$ and $y$", and as you might expect he failed to do so.

Please explain to me how in the world did Kepler guess this relationship working with so few data points. And if my assumption that the number of data points Kepler had at his disposal is small, is wrong, then I still think its quite difficult to guess this relationship without a calculator.

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  • $\begingroup$ In that time scientists were able to make a lot of astronomical calculation by hand (see prosthaphaeresis and logarithmic tables) and Kepler was a very very clever theoretical astonomer. $\endgroup$
    – Mauro ALLEGRANZA
    Feb 11, 2015 at 15:51
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    $\begingroup$ About Kepler's discoveries, you can see at least Alexandre Koyré, The Astronomical Revolution: Copernicus - Kepler - Borelli (original ed.1961). $\endgroup$
    – Mauro ALLEGRANZA
    Feb 11, 2015 at 15:53
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    $\begingroup$ Cross-posted on Astronomy. $\endgroup$
    – HDE 226868 ♦
    Feb 11, 2015 at 16:50
  • $\begingroup$ There's some confusion in the answers below as they seem to suggest that logarithms are an essential part of Kepler's insight. This is not the case. The law states that $P^2 \propto a^3$, which he could have guessed at by looking at the magnitudes of the numbers and easily confirmed by noting that the plot of $P^2$ against $a^3$ is an approximately straight line. Kepler was a very skilled mathematician and was obsessed with patterns, so it is not hard to imagine that he spent a lot of time trying things until he hit on this. $\endgroup$
    – orome
    Sep 10, 2015 at 23:17
  • $\begingroup$ Given that the ancient Greeks spoke of the music of the spheres, it might not be such a great leap that Kepler used a music interval as a power! namely 1.5 the natural fifth. $\endgroup$
    – John Shanahan
    Jan 2, 2020 at 12:37

3 Answers 3

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You can read Kepler's Harmonia Mundi (there is an abbreviated English translation, but it includes that part.) Kepler was looking for all kinds of numerical relations for many years (most of his life). Most of the relations he found in Harmonia Mundi are accidental and are of no value for modern science.

I agree with your friend, that given 6 pairs of numbers (5 planets were known +Earth), after some trials one could discover the 3-d law. (There were more striking discoveries of this sort in history of science, for example Balmer's series, which is even more impressive than the Third law).

Kepler's greatest achievement was the First Law. This was really one of the greatest breakthogh's in the whole history of science. And he described himself in great detail all steps which he made to discover this. Including all mistakes that he made. This description is in his Astronomia Nova. And there is also an English translation, complete! So you can follow each step that he made. If you have time and patience.

Logarithms were just invented when Kepler was writing Astronomia Nova. The third Law was discovered 10 years later. If you plot logarithms of periods against logarithms of distances, you get a straight line. So it would be surprising if someone knowing the data and logarithms, and trying to discover some relation missed this:-)

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I agree that it is amazing and a credit to Kepler's insight into numerical patterns, which is reminiscent of Euler's. It took Kepler extra 12 years to discover the third law after discovering the first two, perhaps exactly because of the comparative scarcity of data points.

According to Kepler, after years of searching for additional patterns on March 8 of 1618 a "marvelous" idea suddenly "appeared" in his head, that "the proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances". In other words, he conceived of what we would call a linear log log fit between mean distances and periods. It didn't fit though... at first, because of a computational error. But on March 15 the idea "came" to him again, and he got the computations right.

A plausible speculation is that Kepler got his miraculous inspiration from reading Napier's 1614 work on logarithms at the end of 1616. It is unlikely that he used graphs or plots but if he converted mean distances and periods into their logarithms using Napier's tables the constant proportionality ratio of $1\frac12$ between the two would have been noticeable. His account suggests however that he first conceived of a pattern in this form, and only then did the computations. He later wrote his own book about logarithms (published in 1621), and used them in computations for Rudolphine Tables.

A word of caution on Kepler's self reporting though. He told a breathtakingly fascinating story of discovering the first two laws in Astronomia Nova. But... "recent research, especially that of William H. Donahue, has shown that the account Kepler offers his readers is not a true history of the course of his research--something Kepler never claimed--but is rather a didactic or rhetorical pseudohistory", see Voelkel, The Composition of Kepler's Astronomia Nova.

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    $\begingroup$ Anyone who does mathematics will be aware of the way in which good ideas can just appear in your head with no apparant train of thought leading up to them. It's very mysterious. Sometimes I think it's worth keeping that in mind when we wonder "how could anyone have come up with this?". If the solution to Exercice 4.1.5 occured to you in a flash of insight, just imagine the kind of nuclear event it must have been when the same thing happened to Kepler or Gauss... $\endgroup$
    – Jack M
    Feb 11, 2015 at 22:47
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    $\begingroup$ Actually Kepler learned logarithms from their co-inventor Jobst Bürgi who worked in his lab. Quoth O’Connor-Robertson: “There is strong evidence that Kepler got the idea for his third law of planetary motion from thinking about logarithms, and it must have been through discussions with Bürgi that logarithms were a common topic at Hradcany. (...) Kepler wrote about Bürgi’s logarithms in the introduction to his Rudolphine Tables (1627). (...)” $\endgroup$
    – Francois Ziegler
    Jun 6, 2018 at 13:39
  • $\begingroup$ Your link to "Voelkel's book" does not work. Please fix it, and add a reference on Donahue work, if possible. $\endgroup$
    – Alexandre Eremenko
    Jan 20, 2022 at 4:32
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Kepler's 3rd law could have been deduced based on the orbital parameters mentioned in Ptolemy's Almagest (of the 2nd century CE). It was not important for this that Kepler had more modern observations of higher accuracy available to him, nor that he had discovered that the orbits were ellipses instead of circles. The following chart shows the orbital sizes (a) and periods (P) deduced from the Almagest as little squares, and shows Kepler's 3rd law as a straight line. The correspondence is good enough to deduce Kepler's 3rd law from the data points.

Kepler's 3rd law from Ptolemy's data

The crucial insight, first obtained by Copernicus (see his Commentariolus, written some time before 1514), was that the heliocentric model of the Solar System allowed for the sizes of the orbits (relative to that of the Earth) to be derived from the available observations. Without this insight, Kepler would not have known the sizes of the planets' orbits, and would not have had the data from which to deduce his 3rd law.

For an inferior planet, closer to the Sun than Earth is, the size of their orbit relative to that of the Earth determines their greatest angular distance from the Sun (their maximum elongation) as seen from Earth. In terms of the geocentric model used in the Almagest, the size of the orbit corresponds to the size of the epicycle of the planet.

For a superior planet, further from the Sun than the Earth is, the size of the Earth's orbit relative to that of the planet determines the size of the planet's opposition loop where the planet shows apparent retrograde motion near its opposition to the Sun. In terms of the geocentric model, the size of the epicycle of the planet corresponds to one over the size of the orbit, because the epicycle of the planet is equivalent to the orbit of the Earth.

Ptolemy gives the sizes of the epicycles as

  • Mercury: 22;30 = 22.5 out of 60 (Almagest, IX 9, Toomer Edition)
  • Venus: 43 + 1/6 out of 60 (X 2)
  • Mars: 39;30 = 39.5 out of 60 (X 8)
  • Jupiter: 11;30 = 11.5 out of 60 (XI 2)
  • Saturn: 6;30 = 6.5 out of 60 (XI 6)

The deduced orbital sizes are, relative to that of the Earth:

  • Mercury: 22.5/60 = 0.375
  • Venus: 43.17/60 = 0.7194
  • Earth: 1
  • Mars: 60/39.5 = 1.5190
  • Jupiter: 60/11.5 = 5.2174
  • Saturn: 60/6.5 = 9.2308

The planet's orbital period follows from their mean daily motion in longitude (for superior planets) or anomaly (for inferior planets). Ptolemy gives the following mean daily motions in anomaly (degrees per day), i.e., motion along the epicycle:

  • Mercury: 3;6,24,6,59,35,50 = 3.106699 (IX 3)
  • Venus: 0;36,59,25,53,11,28 = 0.6165087 (IX 3)

and the following mean daily motions in longitude (degrees per day), i.e., motion along the deferent:

  • Sun: 0;59,8,17,13,12,31 = 0.9856353 (III 1)
  • Mars: 0;31,26,36,53,51,33 = 0.5240597 (IX 3)
  • Jupiter: 0;4,59,14,26,46,31 = 0.08312244 (IX 3)
  • Saturn: 0;2,0,33,31,28,51 = 0.03348854 (IX 3)

For the inferior planets, we have to add the Sun's motion in longitude to the planet's motion in anomaly to get their motion relative to the celestial sphere. The deduced orbital periods in days and years are

  • Mercury: 360/(3.106699 + 0.9856353) = 87.96935 → 0.2408492
  • Venus: 360/(0.6165087 + 0.9856353) = 224.6989 → 0.6151977
  • Earth: 360/0.9856353 = 365.2467 → 1
  • Mars: 360/0.5240597 = 686.9446 → 1.880769
  • Jupiter: 3606/0.08312244 = 4330.961 → 11.85763
  • Saturn: 360/0.03348854 = 10749.95 → 29.43202

These orbital sizes and periods are displayed in the chart.

It might be that the mean daily motions still need to be corrected for precession (said by Ptolemy to work at a rate of 1 degree per 100 years), but that makes very little difference to the chart.

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