# How did Kepler “guess” his third law from data?

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.

• He only used data about Mars. His boss, Tycho Brahe, told him to figure out the retrograde motion of Mars once and for all. And he fantastically did so. The third law came from his astrological pattern fitting in Harmonices Mundi And he had enough data to solve this geometric problem. More data would not have helped him. He actually picked only a subset of the oppositions of Mars that Tycho Brahe had observed. – LocalFluff Feb 11 '15 at 6:59
• Cross-posted on Physics and then migrated to HSM. – HDE 226868 Feb 11 '15 at 16:50
• Kepler had plenty of data to derive his first and second laws, each of which applies to a single planet at a time, but his third law is an entirely different animal. It relates the orbital characteristics of different planets to each other. No matter how much data Tycho had collected, there were only six planets (counting Earth but not counting the Sun or Moon), and their orbital characteristics were not observed so much as calculated (laboriously) by Kepler. Six points, each with a high margin of error, is enough to demonstrate a linear relationship, but barely. – ganbustein Feb 12 '15 at 9:25
• @LocalFluff: I've read too that Kepler basically only used data about Mars. But given that the third law expresses relationships between orbital periods of different satellites, how could he possibly have done that, no matter how much information about Mars alone he had? – Marc van Leeuwen Feb 12 '15 at 13:32
• @MarcvanLeeuwen I think that it comes down to his new physical view of things. That the same set of natural laws universally guide all motions. Others later performed the tedious calculations to confirm this for all planets and the Moon, and Halley for a comet, certainly already in the 17th century. Only the orbit of Mercury didn't quite fit because of subtle relativistic effects. – LocalFluff Feb 12 '15 at 14:14

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.)

• Though I'm hardly a scholar of Kepler's work, AFAIK the attribution of the angels explanation to Kepler is a complete fabrication. Do you have a reference for this that's either written by Kepler or one that directly directly cites Kepler? – Stan Liou Feb 11 '15 at 6:55
• Kepler actually tried to make magnetism (then popular because of William Gilbert) explain the movements of the planets around the Sun. It is this which is the foundation of physical science. He left the angels in church. And he only used selected data about Mars, and had much more data than he could handle. Big Data of his time. Lack of data was not at all his problem. – LocalFluff Feb 11 '15 at 7:09
• Indeed, Caspar p. 67: "It is the new thought that in the sun there is situated a force which produces the planet motions, and which is so much the weaker, the further removed the planet is from the source of the force. To be sure, in his book he speaks of an 'anima motrix,' a moving soul; but already in a letter of this period he uses the word 'vigor,' force." But anima motrix isn't an angel... this German wikipedia article on anima motrix is also interesting. – Stan Liou Feb 11 '15 at 7:15
• @StanLiou Yes, one has to keep in mind the context of the words. "Soul" is a word for force. Just like we today use simple words for natural phenomena and agriculture to describe our technological society: (wheat) field, (fishing) net, (river) current. Even new terms come out as "cloud". We don't mean it literally, nor was the word "soul" always meant literally. A medieval farmer might get quite confused by a textbook about electronics! – LocalFluff Feb 11 '15 at 10:37
• @LocalFluff Yeah, to make a familiar comparison, the original name of kinetic energy was vis viva ('living force'), the term adopted from earlier tradition but does not refer to literal living. The term itself still survives to this day in orbital mechanics, too. – Stan Liou Feb 11 '15 at 10:58

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.

References:

1. Caspar, Max, Kepler, (Dover, New York, 1993).
• Interesting that he mentioned the mean distance. – CodesInChaos Feb 12 '15 at 9:00