How did Kepler "guess" his third law from data?

Question

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.

Answer 1

Kepler's account of how the third law came to be is as follows (Caspar p.286; emphasis mine):

On the 8th of March of this year 1618, if exact information about the time is desired, it appeared in my head. But I was unlucky when I inserted it into the calculation, and rejected it as false. Finally, on May 15, it came again and with a new onset conquered the darkness of my mind, whereat there followed such an excellent agreement between my seventeen years of work at the Tychonic observations and my present deliberation that I at first believed that I had dreamed and assume the sought for in the supporting proofs. But it is entirely certain and exact that the proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances.

Although Kepler does not actually describe inspiration that led him to believe this, the curious phrasing provides a very strong clue when combined with some background biographical information:

1. John Napier published Mirifici Logarithmorum Canonis Descripto in 1614, which contained the then-new invention of logarithms. Kepler was aware of Napier's work by 1617 (Caspar p. 308), perhaps earlier.
2. Joost Bürgi published work on logarithms almost at the same time as Napier, and Kepler was similarly aware of Bürgi, even praising his mathematical abilities as surpassing most professors of mathematics.

Thus, Kepler's statement is is equivalent to saying that the data makes a slope of 1.5 on a log-log graph, a which is a very simple linear relationship on this scale.

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References:

1. Caspar, Max, Kepler, (Dover, New York, 1993).

Answer 2

Kepler's third law is trivial (in my opinion) compared to his first law. I am quite impressed that he was able to deduce that the orbits were ellipses. To get that, he had to go back and forth plotting Mars' direction from Earth and Earth's direction from Mars. He knew the length of both planets' years, so observations taken one Mars year apart would differ only because Earth had moved.

But maybe not so trivial. He published his first two laws in 1609. The third law didn't come along until ten years later, in 1619. With ten years to work on it, even the most obscure relationship will eventually be found.

To discover a ratio-of-powers relationship, plot the logarithms of the numbers. In your example with $x^4 = y^3$, the logs would plot on a straight line with a slope of $3/4$.

The timing is right. Napier published his book on logarithms in 1614. Kepler may have on a whim applied this shiny new mathematical tool to his crusty old data.

The major hurdle was that at the time there were only six known planets, so he didn't have an abundance of data points, and the ones he had were by no means precise.

Kepler's other problem is that none of his laws made any sense to him. They fit the data, but he had no idea why. He didn't have Newton's laws of motion to work from, he had no understanding of force, momentum, angular momentum, and certainly not gravity. So far as he knew, the planets moved the way they did because God decreed it, and angels were tasked with pushing the planets along their orbits. The outer planets moved slower because they were being pushed by lesser angels.

(Feynman makes the comment that we understand so much more now. We now know that the angels are on the outside pushing in toward the Sun.)