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In epicycle-deferent astronomy, adding a second ”minor” epicycle to account for observational discrepancies is observationally equivalent to shifting the deferent into a so-called eccentric, or a circle with a center not at the Earth (see e.g. DeWitt 2010, Worldviews, p. 120, and the illustrations I’ve borrowed from that chapter), as was pointed out already by Hipparchus. Ptolemy famously chose the eccentric approach for the Almagest. I’m trying to wrap my head around how to geometrically prove the equivalence. Minor epicycle

More precisely, the idea I am after is that the addition of a second epicycle to the original epicycloid (compounded only of a deferent and a major epicycle) is supposed to have the same effect upon the movement as projected unto the outer circle of fixed stars by a line of sight from Earth as just shifting the original epicycloid by a distance equal to the diameter of the minor epicycle. I think it’s the earliest proof of an ”empirical equivalence” of two hypotheses. Projection of epicycloid unto the backdrop of the stars.

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I have created an animation to specifically illustrate the equivalence between the eccentric hypothesis and the deferent-plus-epicycle hypothesis. You may view it at EcliptiQc | Eccentric or Epicycle?.

If you read French, I have translated the whole Almagest to French; you may access it by clicking the logo in the above link.

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    $\begingroup$ Thanks a lot! Amazing that you translated the whole work! I do read both French and Ancient Greek, so I can imagine the amount of work you put in... However, the planet is supposed to move on the epicycle (in the opposite direction of the epicycle on the deferent), otherwise one can't model retrograde motion! And I'm unsure how that is achieved with an eccentric; perhaps the center of the eccentric is supposed to move circularly with the same speed and direction as the planet on the epicycle? Anyway, great animation: what software did you use to do it? $\endgroup$
    – Urdatorn
    Jan 4, 2023 at 14:30
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    $\begingroup$ Thanks—It’s JavaScript, HTML, and SVG. The planet actually does move on the epicycle, but it’s not apparent. If it were fixed on the epicycle, it would always lie on the (extended) line from the Earth to the centre of the epicycle… It was only a demonstration of the equivalence between eccentric and deferent-and-epicycle: they didn’t consider retrograde motion (yet), which would be a downfall of early Greek planetary theory. As for translation, I started from Toomer’s English translation and an old French translation by Halma, with some reference to early manuscripts for disambiguation. $\endgroup$ Jan 5, 2023 at 1:50
  • $\begingroup$ Thanks for your answer! What do you mean by downfall? As in the picture above, the magic of epicycles is that they model retrograde motion. $\endgroup$
    – Urdatorn
    Jan 6, 2023 at 23:11
  • $\begingroup$ Yes, it does model retrograde motion, IF the planet does NOT move in the same way as in my animation, but the first expressions of the model were along the lines of my animation, hence why they didn’t explain retrograde motion. $\endgroup$ Jan 7, 2023 at 2:23
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Following the comments of @Pierre Paquette's answer, I think it worthwhile to clarify several things up:

  • The "geometrical equivalence of hypothesis" that is mentioned in the question is not related to the minor epicycle. Rather it is about the equivalence of an eccentric to a concentric with one major epicycle.

  • In The model with the epicycle, the planet is moving on the epicycle in the same speed the epicycle itself moves but in opposite direction. so it keeps the same absolute angle with respect to the center of the epicycle(in @Pierre diagram it always on the "top").

  • To show the equivalence is relatively easy geometrically by Congruence of triangles. (just make sure the that eccentricity is equal to the radius of the epicycle.).

  • That was a simple model that was not actually used for the planets but only for the Sun. To this provisional model, an equant was added, which made the movement of the epicycle to be of unequal speed.

  • But even with equant, that model was not sufficient to represent the movements of the planets in the Sky since static-earth required another epicycle to account for the retrograde motion (With Ptolemy eccentric model which was lacking the major epicycle, this minor epicycle is actually the only epicycle in the model). The way to eliminate this minor epicycle would be by having the Earth moving (basically the movement of Earth around the Sun is like the movement of the planet in the minor epicycle).

  • Now, the critical reader might know that even Copernicus - that had the Earth moving in his model - has used two epicycles. Given the above, we would ,however, rather expect only one major epicycle in his concentric model, since the retrograde motion did not require another epicycle in his model. Yet, Copernicus did use another small epicycle to account for the "equant" which was missing in his model: with a correct ratio between the radii of his two epicycles he could achieve about the same (but not totally equal) movement as a Ptolemy model with eccentric equant and one minor epicycle.

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