Kepler’s third law is so simple to state that (as shown last time) it is something that any grade school kid, armed with Copernicus’s data and a calculator, can verify. Yet it was 75 years from Copernicus’s publication til Kepler discovered this formula! Why did it take Kepler until 1618, nearly 50 years of age, to recognize such a simple relationship? Were people just dumber than high-school students back then?
Here’s a clue. We take all sorts of math for granted that didn’t exist four hundred years ago, and calculations which take an instant now could easily take an hour or even all day. (Imagine computing the cube root of 4972.64 to part-per-million accuracy by hand.) In particular, one thing that did not exist in Copernicus’ time, and not even through much of Kepler’s, was the modern notion of a logarithm.
Although the idea of logarithms of simple numbers (such as logn np = p where n and p are integers) was clearly known to a number of historical figures, including even Archimedes, and including several Islamic scholars, the idea that they could be applied to any number, and serve as a practical tool to simplify calculation (as in the slide rules that were in every engineer’s and scientist’s pocket before electronic calculators), seems to have been the invention of John Napier. Napier, after decades of work, published his method in 1611. Here’s an image, from Napier’s book, of what is quite possibly the first-ever plot on a logarithmic axis.
Prior to this time, it was probably very hard even to imagine actually transforming data so it could be considered on logarithmic axes, and there was no method for calculating logarithms for arbitrary numbers. But this transformation turns something subtle and hidden into something obvious, as we saw in my last post.
Kepler soon learned of and studied Napier’s new methods, even writing documents significantly advancing the subject in 1621 and 1624. (It turns out that Joost Bürgi, whom Kepler knew well, apparently understood precision use of logarithms too, but didn’t pursue it as far as Napier until after Napier’s book had appeared.)
The absence of logarithmic math, combined with the many possible choices for the distance R (see the last post), presumably explains why Kepler, and Copernicus before him, didn’t easily stumble on the right answer earlier. Remember, calculating squares and cubes and square-roots and cube roots precisely was very hard work, and Kepler didn’t trust a calculation that wasn’t precise.
But once logarithms could be calculated, they made complicated power laws (such as R=T2/3 or equivalently T=R3/2) much easier to spot. (If the needed power law had been a simple square or cube, I wonder if Kepler and perhaps Copernicus might have noticed it earlier.) Evidence that logarithms were central for Kepler’s breakthrough is found not only in the timing but also the wording he used to describe his discovery: he phrases it as a fractional power law. [“The proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances.”]
This is an example of how mathematics can spur new calculational techniques, and these in turn can lead to physics breakthroughs. It’s one reason that many physicists learn a lot of mathematics, and why they may even spend part of their careers developing it. Newton’s and Leibniz’s inventions of calculus made it possible to solve all sorts of previously impossible problems (such as the proof that planets subject to Newton’s law of gravity have elliptical obits.) What mathematicians achieved in studying curved geometry in the 19th century eventually made its way into Einstein’s theory of gravity. The mathematical theory of fiber bundles and more general issues in topology turn out to be crucial in understanding important subtleties in quantum field theories, some of which have direct experimental consequences. So even though physics is unquestionably an experimentally-founded subject, one should be cautious in dismissing a physicist’s work as overly mathematical and disconnected from the real world. Sometimes the mathematics is just what is needed, and it’s often very hard to see that in advance.