And so for the curves in question, the Fourier expansion would have only a finite number of terms.
The point being that, in contrast to what was being asserted, Ptolemy’s concept is subsumed within the modern one; the modern language is more general, capable of expressing not only Ptolemy’s thoughts, but also a heck of a lot more. In effect, modern mathematical physics uses epicycles even more than Ptolemy ever dreamed.
This is false in an amusing way: expressing motion in terms of epicycles is mathematically equivalent to decomposing functions into Fourier series—a central concept in both physics and mathematics since the nineteenth century.
To be perfectly fair, AFAIK Ptolemy thought in terms of a finite (and small) number of epicycles, not an infinite series.
And so for the curves in question, the Fourier expansion would have only a finite number of terms.
The point being that, in contrast to what was being asserted, Ptolemy’s concept is subsumed within the modern one; the modern language is more general, capable of expressing not only Ptolemy’s thoughts, but also a heck of a lot more. In effect, modern mathematical physics uses epicycles even more than Ptolemy ever dreamed.
That’s good point; I haven’t thought about that. Go epicycles ! Epicycles to the limit !
ducks and runs away