Who Determined the Phase and Period of Epicycles in Greek Astronomy?

[puncheex]

I've researched multiple histories of Greek astronomy, and have not found a satisfactory answer to this. When describing multiple deferents, each with a first order epicycle, many illustrations show the epicycles at different phases of their rotations: Mars will be at 37 degrees, Jupiter at 123, Venus straight up at 0, Saturn at 280. Other illustrations have them all in lock-step, all at, say, 190 degrees and all rotating (if it is an animation) in sync. In fact, the deferents of the sun, Mercury and Venus around the Earth are also synced to them, to keep them near the sun. This is, of course, the correct way of showing them, the only way that works, because these first level epicycles and the three interior deferents (not the moon's!) all must be in sync with the sun and therefore have periods of exactly a year, because they all represent the correction brought to Ptolemy's model by placing the Earth at the (near) center of the solar system.

The question is, who determine that the first order epicycles had all to be equivalent in phase and period? Eratosthenes? Ptolomy? The results that Ptolemy produced are proof that his were that way, but the Almagest treats each planet/Earth pair as being entirely independent in all ways from all the others (in fact, he uses a normalized deferent and appropriate ratios rather than hard numbers) EXCEPT that all the epicycles were in phase. Does anyone know who invented that fact, and what the justification for demanding it was? A citation in the literature would be welcome.

 

[puncheex]

Well, actually I managed to find the answer. For those who are interested...

Part of the question was incorrectly stated. I said all the epicycles have a common period and phase (they all point the same way). That's true only for the epicycles of the superior planets (Mars, Jupiter, Saturn). For the inferior planets (Mercury, Venus), it's actually the deferent which matches period and phase. They do so because of constraints placed on the system by observation.

The superior planets reach the midpoint in their retrograde only when a straight line connects the epicycle center to the planet and also the sun and the Earth. As it does this at every retrograde, the epicycle has to rotate at the same speed as the Earth-Sun line. If it does so for all the superior planets, then they are all in sync with the sun orbit.

For the inferior planets, it is the fact that the center of each planet's deferent must be on the sun-Earth line because they are observed to only excursion an equal amount to each side of that line periodically.

So, the sun-earth line (sun deferent, if you will), is in sync, both period and phase, with the deferent of the inferior planets and the epicycle of the superior planets. It is interesting that Ptolemy never states the sizes of the deferents or epicycles in the Almagest (though he does essay guesses in the Planetary Hypotheses), but only computes ratios between the deferent and epicycles radii. If it had happened that he could get good sizes for either, then he would have found that all these orbit components which have identical periods and phases would also have identical sizes, all equal to the sun's deferent. Had he ignored that red flag, he'd really win the crown as absent minded professor.

 

[bitznbitez]

Thanks for the info. I've been fascinated by the epicycle model, not for its correctness but for its ability to do a reasonable job describing the observations of those days.

 

[Chronos]

The epicycle model was originally devised by Apollonius in the third century B.C. - http://abyss.uoregon.edu/~js/glossary/ptolemy.html.