Could epicycles approximate anything to any precision? In what way is QED different? Just curiosity, as follows: I was trying to explain/illustrate to a non-technical friend that physics is just "mathematical models", which may or may not represent/correspond_to some "underlying reality". And we can't infer it does just because the calculated numbers work (correspond to observed measurements).
And the example that crossed my mind was this: classical Greeks (mostly) thought planets revolve around the earth. But to explain retrograde motion, etc, they introduced epicycles. And when that didn't exactly work, they introduced epicycles on epicycles. Now, I suggested, if they'd known a little more math, they could've "expanded" the observed motion in epicycles (if epicycles are "complete" for describing such orbital curves). And then they could've argued along the lines, "Look, our calculated numbers are accurate to 16 significant decimal digits. So our epicycle model of planetary motion must be right. How could we obtain such incredible accuracy otherwise???"
So how good/bad an illustration is this? Are epicycles complete in this sense? And, of course, an underlying question I didn't mention to my friend: how can you "protect" QED, etc, from such objections? Or can't you?
 A: Yes they could, here is a fun YouTube video where Homer Simpson is sketched by a tower of epicycles. We can think of epicycles as sums of $a_ne^{i\lambda_n t}$ by identifying the plane of the ecliptic with the complex plane. The $a_n$ then code their radii and phases and $\lambda_n$ are the frequencies. This is a generalization of the Fourier expansion, in which $\lambda_n$ must all be integers. Any continuous almost periodic function admits such an expansion, the epicyclic exponents are complete in their space (some discontinuous functions can also be approximated). Such functions were introduced and studied by Harald Bohr, the famous physicist's brother. In particular any continuous periodic motion can be accomodated by epicycles. For more details see Mathematical Power of Epicyclical Astronomy by Hanson.
As for QED, it is at least protected from the Homer Simpson objection, unlike epicyclic astronomy it only has finitely many parameters to "adjust". And it made predictions of (even today) remarkable accuracy without any analog of mounting epicycles upon epicycles that Islamic astronomers engaged in at the end of middle ages, see Ancient Planetary Model Animations, especially the Arabic models for outer planets.
A: A mathematical model that could explain anything, doesn't actually explain anything.   
The trouble with epicycles is that they can be used to fit anything.  See the Wikipedia article on epicycles.  Newton's theory of gravity is much better because the orbits that it predicts must be ellipses (in the first approximation) and the rest of Kepler's Laws also follow.  
As with a lot of questions of this sort (the philosophy behind physical reasoning) I suggest you check out Richard Feynman's lecture on physics.  
