Epicycles on Epicycles
Although the Ptolemaic system is often condemned in the history and science books on the basis of Occam's razor, for having required the heaping of epicycle on epicycle to approximate the motion of the stars, it's recently occurred to me that I've yet to see anywhere a demonstration that the problems with such an approach don't go any deeper than that. Is it really the case that with enough epicycles, one can approximate the elliptical motions of the planets to an arbitrary level of accuracy? Or, in mathematical language, do epicycles constitute a basis for a space in which all celestial motions can be shown to be embedded, in an analogous manner to that in which sine and cosine functions constitute a complete set for Fourier series?
My suspicion is that the Ptolemaic system of epicycles would indeed have been workable had the ancients enjoyed the facility with infinite series we do today, and in fact I have good reason to believe that the Ptolemaic epicycles-on-epicycles are little more than a disguised version of Fourier analysis, the key item linking the two theories in my mind being the function eix. While I don't have time to work out the mathematics in detail, I don't imagine that it should be all that difficult to do: the essential thing to keep in mind is that the motions of the planets are periodic, and as such, ought to be amenable to the methods of harmonic analysis.