How did pre-Copernican astronomers accurately predict planetary position? Copernican elements (circular orbital elements) are not very accurate. But Copernicus simplified our understanding a great deal by placing the Sun at the center of the system.  Im astonished by the historical accounts that, say, Mayan or Babylonians, or whoever may have had the ability to predict planetary position an accuracy that competes even with Kepler.  Now, Im only curious about the reasoning and the mathematics behind making such predictions from pre-Copernican understanding.  What approach did these ancient people take?  I cant seem to find any online reference to this approach.
 A: The excitement behind various claims is somewhat excessive.
First, the Mayan astronomers, see e.g. Mayan astronomy at this page, didn't use any armillary spheres or sextants as others did. Their observations were made with naked eye and they were depicting positions of planets with crosses. The accuracy of the Venus' position after a synodic 584-day cycle was off by 2 hours.
It's good but it isn't unbelievably accurate in any sense. It just means that the moment when the Venus "returns to the original place" was measured with a 2-hour accuracy. As far as I know, all the Mayan models for the orbits were periodic and the imagined trajectories were circular. They have just played with lots of periods which is why they ended with lots of these baktuns and pictuns (cycles) and parallel calendars. But it was numerology based on the known periodicities, not really a precise framework to predict the positions. Let me also emphasize that there was no "dogmatism" concerning the identity of the "center of the Universe".
The Babylonians did all the astronomy with the tables. Lots of tables. Of course, they could do things much more accurately than the Mayans. It was a phenomenological approach with a lot of hard work whose basic logic only depended on arithmetics applied to measured data adjusted so that the agreement with all the data is reached. They were mapping the trajectories of the Sun and the planets in some coordinates, observed some periodicities, and were ready to calculate all the required corrections by some arithmetic formulae that were simply a good fit.
They didn't have any prejudices about the circular shape of any orbits, centers of the Universe, relationships between planets and cosmology (birth of the Universe), and so on. It was a heavily empirical approach. You could say that it was modern science but you would be missing the fact that they were simply not looking for the laws of physics that explain all the patterns and they were not attempting to reduce the amount of the independent mess. The mess was immense, the trajectories looked rather general. Effectively, the success and precision of their predictions for the locations boiled down to their having very complicated "laws of physics", laws with tons of (measured) corrections and special rules for individual planets and individual situations.
That's why the Greek astronomy represented dramatic progress because they were actually trying to make things simple, to see a rational explanation of the causes behind the particular trajectories. They were not satisfied with the empirical side of the story and that's why it was such a breakthrough. They were building mechanical models. When it comes to the accuracy, whether the orbits were thought of in the geocentric or heliocentric frame is an irrelevant technicality. The heliocentric frame is closer to allowing us to write Newton's equations as the full classical explanation which is a "culmination" of non-relativistic celestial mechanics. However, one has to do lots of other things before these final steps in order to represent the observed positions on the two-sphere by some actual trajectories in the 3D space which have a cause.
So the Babylonian models were "overfitted", using an immense amount of observed data that weren't independent from each other. Greeks wanted simple enough models that account for the facts. Circular motion is the most symmetric one so they started with it and tried to match the observations in various ways. It didn't work quite exactly so they began to build the epicycles (smaller rotating circles) on top of the deferents (main circles that carry the objects).
Epicycles are often talked about in a negative light but they represented (and similar methods represent) a very progressive intermediate phenomenological description in between the "Babylonian tables" and "simple models of trajectories given by simple equations". The idea that one circle carries another which carries the planet is pretty much equivalent to some kind of a Fourier expansion of $\vec r(t)$ and the Greek astronomers were picking several leading coefficients of this expansion (by observing them). I am sure that most people who talk dismissively about epicycles wouldn't be capable of doing anything of the sort – and anything superior, either.
Ptolemy's model was finally a very accurate description. It was a geocentric model – not only the Sun but also the other planets (and, correctly, the Moon) – were directly orbiting around the Earth. But that didn't hurt the accuracy because the orbits were described in such a way that the Sun-Earth separation was correctly subtracted, anyway.
Now, note that if the exact distances of the planets were known one could notice that in the case of all planets, we are suspiciously subtracting the same periodic function with the period of 1 year, namely the Sun-Earth separation, and we could figure out that in the heliocentric frame, things get more natural. However, when you don't know the absolute distances, things are a bit tougher. They were not aware of the possible task "get simpler by going heliocentric" so they didn't pursue it. It wasn't a problem. Even in Newton's physics, we may describe things in the Earth's frame (even a spinning, non-inertial frame) as long as all the fictitious forces are correctly added.
It's another myth when people believe that the heliocentric model implies that it has no epicycles. Instead, the Copernical model had an even greater number of epicycles than Ptolemy's model because some of the new ones replaced the equants. Ptolemy's equants were actually a rather clever observation that played a similar (although not quite as accurate) role as Kepler's observation that the Sun sits at a focus of the elliptical orbits. Of course, Kepler's precise observations about the ellipses were not known to Copernicus and they don't directly follow from heliocentrism.
A: Luboš Motl addressed the reasoning very well, but for the mathematics:
Richard Fitzpatrick's free e-book A Modern Almagest: An Updated Version of Ptolemy’s Model of the Solar System presents a modernized version of Ptolemy's Almagest.
The following classic paper shows how epicyclic astronomy can be re-expressed in the modern mathematical idiom of complex Fourier series, something that is more familiar to us moderns:
Hanson, Norwood Russell. “The Mathematical Power of Epicyclical Astronomy.” Isis 51, no. 2 (June 1960): 150–58.
See the related video:Ptolemy and Homer (Simpson), 2008. http://youtu.be/QVuU2YCwHjw.
