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I've come across this diagram of the Ptolemaic model. Am I understanding correctly that this means that all the planets' deferents rotate around a point which next to Earth, while Venus' deferent rotates around an epicycle rotating around the same point. Is that correct?

enter image description here

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    $\begingroup$ Where is the diagram from? There were multiple versions of the Ptolemaic system circulating in the middle ages. I cannot make out what goes on in the middle, but eccentrics (centers of the deferents) were typically different for different planets, and, at least in the original version, there were also equants, points from which rotations appeared uniform, see mathpages. Regiomontanus had rotating eccentrics for Mercury and Venus, so, possibly, this is his version or a variation thereof. $\endgroup$
    – Conifold
    Commented May 28, 2023 at 7:44
  • $\begingroup$ I can't remember where I found this, but your answer clarifies a lot. If you add it as an answer I can mark it as the official one. Thank you very much! $\endgroup$ Commented May 28, 2023 at 10:32

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There were multiple versions of the Ptolemaic system circulating in the middle ages, including those where Venus did have a rotating eccentric, the center of the deferent, e.g. in Regiomontanus's Epitome of the Almagest. Ptolemy himself claimed (mistakenly) in Almagest XII.1 that such a model cannot be used for inner planets, which is puzzling because a simple parallelogram construction shows that an epicycle can always be interchanged with a rotating eccentric. Duke has a helpful list of medieval Ptolemaic models with animations, but, unfortunately, the animations require Adobe Flash and do not work anymore.

The original version of Ptolemy's Almagest is described e.g. by Duke in Ancient Solutions of Venus & Mercury Orbits, and its mathematical details can be found on mathpages. Eccentrics were typically different for different planets, and there were also equants, points from which rotations of the epicycle centers on the deferents appeared uniform.

Many Islamic astronomers were dissatisfied with eccentrics and equants for philosophical reasons and replaced them in their versions by additional epicycles, see Islamic astronomy by Gingerich:

"The equant in particular was objectionable to philosophers who thought of planetary spheres as real physical objects, each sphere driven by the one outside it (and the outermost driven by the prime mover), and who wanted to be able to construct a mechanical model of the system... Furthermore, the equant violated the philosophical notion that heavenly bodies should be moved by a system of perfect circles, each of which rotated with uniform angular velocity about its center. To some purists even Ptolemy's eccentric deferents, which moved the earth away from the center of things, were philosophically unsatisfactory.

[...] Al-Tusi found the equant particularly dissatisfactory. In his Tadhkira ("Memorandum") he replaced it by adding two more small epicycles to the model of each planet's orbit... Finally a completely concentric rearrangement of the planetary mechanisms was achieved by Ibn al-Shatir, who worked in Damascus in about 1350. By using a scheme related to that of al-Tusi, Ibn al-Shatir succeeded in eliminating not only the equant but also certain other objectionable circles from Ptolemy's constructions."

European astronomers of late middle ages/Renaissance, like Regiomontanus, inherited such modifications and made variations of their own. Ibn al-Shatir's and Regiomontanus's modifications were instrumental in helping Copernicus transform geocentric models into mathematically equivalent heliocentric ones (although exact transmission of their ideas to him remains controversial), see Swerdlow, Copernicus’s Derivation of the Heliocentric Theory from Regiomontanus’s Eccentric Models.

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