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A Japanese book on mechanics says the following (English translation by me):

As a young boy, Tycho Brahe became interested in astronomy when he was impressed by the fact that the solar eclipse occurred almost exactly as predicted. Later, faced with the fact that a prediction of the celestial proximity of the two planets based on a heliocentric model was off by several days and a prediction based on a geocentric model was off by a month, he realized the need for precise observation of celestial bodies.

『物理学序論としての力学』(藤原邦男)

I would like to know how the prediction performance of the Copernicus and Ptolemaic models compare.

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The Ptolemaic and Copernican models are observationally indistinguishable, so the book must be misunderstanding the question or summarising it inaccurately.

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    $\begingroup$ Unfortunately, this answer cannot be correct. The tables of Ptolemy and the Alfonsine Tables used the same Ptolemaic model, with some different parameters. So their predictions were not the same. Likewise for the different details in Copernicus' 'De Revolutionibus' and in Reinhold's Prutenic Tables (also Coperican in model). There are no grounds for saying that the predictions of these tables were 'observationally indistinguishable'. The answer unfortunately confuses between prediction performance and certain gross features of the models that would be difficiult to tell apart. $\endgroup$
    – terry-s
    Oct 26, 2022 at 7:12
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To elaborate Martin's answer slightly, the Copernican (heliocentric) model is indistinguishable from the Ptolemaic (geocentric) model in its final form.

This is because both models in their final form are actually mechanically equivalent.

What distinguishes planetariums/orreries based on each model is that the geocentric planetarium attempts to explicitly describe the movement of the heavens as seen from Earth (so that things like retrograde motion are explicitly seen in a geocentric planetarium).

Whereas in the heliocentric planetarium, the movement of the Earth together with the movement of other elements makes it more difficult to infer the observational reality we see from Earth.

The heliocentric planetarium effectively demonstrates things as would be seen by a person standing on the surface of the Sun - which is not an observation that has ever been made in practice.

Nowadays, the heliocentric model is regarded as the "simpler" one, but in fact the central importance of Earth to humanity, and the fact that historical interest in astronomy was primarily driven by a desire to model what was seen in the sky (not to represent some abstract cosmological view), meant that for a long time the geocentric model was regarded as the simpler and more obvious description.

It's also important to note that both models actually date from antiquity - the heliocentric model doesn't actually originate with Corpernicus, but is at least as old as Ptolemy who was also familiar with it.

What probably harmed the geocentric model most was that improvements in telescopy meant that irregular orbits and patterns of eclipses were more readily seen and precisely quantified, which increased the mechanical complexity of geocentric planetariums that accounted for them. Better telescopes also meant that an increasing number of elements had to be accounted for.

The "equants" and "epicycles" and so on of the geocentric model, which exist to represent very specific motion which is apparent/relative to Earth, but are unreal in the sense that they do not represent a general description of astronomical movements, massively increases mechanical complexity, and as more elements are added there is a risk that the sweeps of moving linkages and supports start to conflict with each other.

It was also not entirely clear in the past whether such apparent movements in the heavens were "real" or not, but the increasing complexity and arbitrariness of the geocentric model over time, led many to doubt. In that sense, there is a certain philosophical side to the geocentric model which has been falsified - the question of whether it describes something "real", or whether it only describes what we see.

Eventually designing such machinery to represent the geocentric model became too much, so that the geocentric approach began to fail in its own terms (of providing a complete description of the observational reality from Earth, in the form of some conceivable machine that demonstrated the model in motion).

By contrast, a planetarium that implements the heliocentric model is mechanically very much simpler, and the movements of the planets relative to the Sun are seen to be very much simpler and more regular, even though relating it to any earthly observation involves additional reasoning.

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    $\begingroup$ This is not true: Copernicus and Ptolemaic models are not equivalent. The difference between them is not just the difference between heliocentric and geocentric models. The major difference is how these models take into account deviation of planets motion from uniform circular motion. Ptolemy used equant and Copernicus used small epicycles. Ptolemy's equants and Copernicus' epicycles were used not to represent motion relative to Earth, but the deviation of real planet motion (now known as Kepler motion) from uniform circular motion. $\endgroup$ Nov 1, 2022 at 18:34
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    $\begingroup$ The answer would benefit from citation to actual contemporary documents -- both to confirm and to clarify the points being made, which are left obscure in the absence of explanatory documents. $\endgroup$
    – terry-s
    Nov 1, 2022 at 22:39
  • $\begingroup$ @AlexeiKopylov, just to clarify, when I refer to the "Ptolemaic model", I mean a geocentric model which was constantly innovated over the centuries but which maintained the basic principle of the Earth being central and stationary, I don't mean a model written by Ptolemy's own hand. My understanding is that in its latter form, this geocentric model basically contained enough additional mechanical terms (including unreal elements like equants and deferents etc.) that it became an implicit heliocentric model, but applied so as to represent the picture seen directly in the Earth's sky. (1/2) $\endgroup$
    – Steve
    Nov 2, 2022 at 10:18
  • $\begingroup$ Obviously, there must be a genuine mechanical correspondence between what we actually see in the sky from a geocentric perspective, and the heliocentric model, so we do not need to wonder whether it is possible in principle to devise a model which shows everything as moving around the Earth. (2/2) $\endgroup$
    – Steve
    Nov 2, 2022 at 10:18
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    $\begingroup$ It has been known for a while that, from a modern perspective, Ptolemy's model corresponds to the representation of the apparent orbits as Fourier series, see Norwood Russell Hanson's 1960 paper and a visualisation on YouTube. $\endgroup$
    – Tom Heinzl
    Nov 3, 2022 at 9:57

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