# Has anyone explored Ptolemy's epicycles as an early form of Fourier analysis?

Whilst researching science in the ancient world, I came across an observation, which unfortunately I did not make a note of, and so cannot credit, that Ptolemy's epicycles were an early form of Fourier analysis. This made immediate sense to me. After all, we know that smooth functions can be arbitrarily well approximated by Fourier series. Hence any motion can also be.

Has anyone actually made a study of Ptolemy's epicycles from this angle?

• Ptolemy himself never put epicycles on top of each other, so his astronomy did not involve Fourier approximations even implicitly. His Islamic successors did do that, especially Ibn-al-Shatir. For a Fourier interpretation of chained epicyclic models see Hanson, The Mathematical Power of Epicyclical Astronomy. Nov 19, 2020 at 10:00

As @Andrei Kopylov noticed, epicycle theory is not the theory of Fourier series of a periodic function. Still this is called (generalized) Fourier analysis. Such functions are called almost periodic or quasi-periodic, and they expand into generalized Fourier series of the form $$\sum c_k e^{i\lambda_k t},$$ with arbitrary real $$\lambda_k$$, which is also called a (generalized) Fourier series. From the modern point of view such series occupy an intermediate position between the ordinary Fourier series and Fourier integrals. But the general idea is the same: to represent a motion as a superposition of uniform circular motions, and the idea was clearly stated for the first time in ancient Greece. Mathematical theory of such series was systematically developed only in 20 century (H. Bohr, H. Weyl, N. Wiener, ..., A. Kolmogorov, V. Arnold...). A historical survey can be found here:

Quasi periodic motions from Hipparchus to Kolmogorov

Remark. In the earliest models of Apollonius and Ptolemy the periods were commensurable. The first person who clearly realized that the motions of planets have to be modeled with non-commensurable periods was Nicole Oresme (1320-1382).

• An interesting answer. What was the source for the idea that in Ptolemy's earliest models the periods were commensurable? And which work of Oresme do you refer to -- thanks. Dec 29, 2020 at 13:09
• @terry-s: 1. This idea comes from reading Ptolemy and exposition of his work by Neugebauer. 2. The references on Oresme and discussion of his work can be found here: mathoverflow.net/questions/269893/… Dec 29, 2020 at 15:40

Yes. That was done by Giovanni Schiaparelli in a book called Scritti sulla storia dell'astronomia antica, published in 1926 and reprinted in 1997.

This is very popular myth, but it is not true: Ptolemy's epicycles are not Fourier analysis! Fourier series can indeed approximate an arbitrary periodic function. And you can approximate an arbitrary motion of period $$T$$ by the series of epicycles. The first is just a circular motion of period $$T$$, the second is an epicycle with period $$T/2$$ and so on. But apparent planetary motion is not periodic: it is sum of two motions with different periods: one is proper planet motion and another is Earth motion. So Ptolemy's epicycles account just for these two motions. It's not Fourier approximation.

However because planets move not uniformly by circles but by Kepler's laws, the simple model of epicycles would give a very rough approximation. Here Fourier approximation would help: you can add additional epicycle to account for that. But Ptolemy's system did not do that. Ptolemy is most famous for inventing the model of equant. In this system a planet (or actually a center of epicycle) is moving on circle but not with uniform speed, it's moving with uniform angular speed with respect to some imaginary point that lies off center.

But there is some grain of truth in this myth. People indeed used epicycles that you would get Fourier's analysis of planetary motion. But it was done not by Ptolemy, but by... Copernicus. We know Copernicus for inventing heliocentric system. Because of this system Copernicus did not need Ptolemy's epicycles. But this is only one part of Copernicus work. The part that Copernicus was most proud was that he abolished Ptolemy's equant, but introduced new epicycles instead. These new epicycles were exactly the ones that we would get from Fourier's analysis. They have period $$T/2$$ as you would expect. This system had the same order of error as Ptolemy's: Ptolemy equant system had error $$\approx 1/2e^2$$, and Copernicus epicycles had errors $$\approx 3/2e^2$$ where $$e$$ is eccentricity, but Copernicus system was a little bit simpler, because it was sum of two uniform circular motions.

So, to recup: Copernicus got rid of Ptolemy's epicycles (which are unrelated to Fourier's analysis), because he put Sun in the center of Universe. But introduced new epicycles (which are the first Fourier approximation) instead of Ptolemy's equant system (which had the same purpose: approximate unknown Kepler's motion).

BTW, another legend that usually goes with this myth says that people before Copernicus added more and more epicycles to Ptolemy's system to make it more accurate. This is not true. No one before Kepler made a more accurate system that Ptolemy's. Fourier's epicycles was indeed added by Copernicus, but not to improve accuracy, but to make it simpler. And it was just first approximation (if we consider uniform motion as zeroth approximation). There was no epicycles that correspond to second term in Fourier series.

• Mounting multiple epicycles on top of epicycles was done before Copernicus by Islamic astronomers, e.g. by Ibn-al-Shatir, who might have been Copernicus's source. Nov 19, 2020 at 10:33
• Yes. This is true. But Ibn-al-Shatir's system is equivalent to Copernicus system, but in geocentric settings. Ibn-al-Shatir replaced Ptolemy's equant by "Fourier" epicicle. But did not get rid of Ptolemy's epicycle. So indeed Ibn-al-Shatir's system had epicicle on epicicle, but only one epicycle corresponds to Fourier term. And it was not added to make system more precise, but to make it more simple. Nov 19, 2020 at 16:37