### 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.

Cross-posted on Physics and then migrated to HSM.

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.

@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?

@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: That does not answer my question at all. One _cannot_ compute the slope from a graph having only one data point (the orbit shape and orbital period of Mars), no matter how precise that data point is.

@MarcvanLeeuwen He had the orbit of Earth too. But one single orbit is enough to test the third law, and then it is a philosophy of physics to generalize that result.

7 years ago

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?

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.

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.

@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 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.

When you say Kepler didn't have an abundance of data points, I'm not sure that's strictly true. Tycho Brache had better equipment than anyone before him, so the planetary data points Kepler got were 10 times more accurate than people had previously. He also had Copernicus' data, form which he was able to deduce that Mars orbit was almost 2 degrees off earth's elliptic (something Copernicus hadn't noticed). With better data, the first person to recognize different elliptic planes and a solid mathematical foundation he was able to work things out that nobody else had before.

I'd post a link to back that up but I can't find it right now. You're right that Kepler's laws didn't make sense to him though. I've read that too. He actually didn't like ellipses and he didn't understand why orbits weren't circular. He liked circles better, but couldn't make them match the data.

Great answer. Would you please tell me how Kelper know Mars's planet year?