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Could ancient astronomers have proven heliocentrism and, if so, how could they have done so?

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You can't ever "prove" heliocentrism! (We can't even today. Thinking that science "proves" things is not at all helpful in doing science.) What we can do is show that a heliocentric model of the Solar System is simpler and explains more observations with fewer assumptions. (E.g., a heliocentric model is compatible with Newtonian mechanics and a world where earthly mechanics and heavenly motions are explained by one theory is very persuasive compared with older models of planetary motion. But it is not proof.)

The ancient Greeks and the medievals ran into the same basic problem: They were convinced that motions in the sky must be circular and, in the absence of Newton's work, circular motions around the Sun were no more elegant and no more explanatory than circular motions around the Earth. (One of the main reasons Copernicus' theory was not quickly accepted is that it used circular motion and thus made seriously wrong predictions of the motions of the planets, especially Mars. By around 1600 we had excellent naked eye measurements of planetary motion, probably far better than what the Greeks had. It was Kepler's elliptical orbits which made heliocentrism convincing.)

Remember also that measuring absolute distances to celestial bodies was very difficult without telescopes, so the Sun being enormously more massive than the Earth was not obvious, making its centrality also less obvious.

It's also worth remembering that the focus of astronomy up until the early 1600s was on understanding and predicting the motions of the planets, and not understanding them as physical bodies. (Remember that even in the 1800s some reputable scientists doubted that we would ever be able to understand the true nature of the planets and stars -- then spectroscopy was developed.)

On the other hand, the Greeks knew about ellipses, so a clever Greek mathematician could have proposed elliptical orbits (for Mars and the Moon, at least) and come up with a theory which predicted their motions very accurately. But until you know the relative masses of the Sun and Earth or start to have an inkling of Newtonian dynamics and gravitation, it's really not terribly important which body is at rest.

Bottom line: There was just too much science yet to be developed for the Alexandrian Greeks to find a heliocentric Solar System to be compelling. (But even then, never proven.)

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    $\begingroup$ Yes we can. The deep space probes would not have worked if geocentric theorem were true. $\endgroup$
    – Joshua
    May 29, 2023 at 0:52
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    $\begingroup$ @Joshua Mark is making a general, philosophical point about how science works. You can't really ever prove something outside mathematics. You can disprove things, so you can argue that the probes' working disproves geocentrism, but that doesn't prove heliocentrism. $\endgroup$
    – terdon
    May 29, 2023 at 11:01
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    $\begingroup$ Before Newton there was another strong argument against heliocentrism: why do things have weight? The accepted model was that all matter has a tendency to strive to move towards the center of the world. But if that center is not the center of the Earth, then what keeps things striving towards the center of the Earth? (sadly, due to certain unwise political decisions made by Galilei, the popular consciousness mistakenly views the debate of his era as one between science and ignorance, but before Newton, heliocentrism wasn't all that obviously better at explaining how the world works) $\endgroup$
    – vsz
    May 29, 2023 at 16:57
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    $\begingroup$ To not open a new answer just for this, I'd add stellar parallax to your "bottom line". Stellar parallax was first successfully measured in the 19th century, and that was definitely a very strong evidence for heliocentrism versus geocentrism. And 19th century manufacturing and precision was way more advanced than anything the ancient astronomers could have managed. Stellar parallax is also important because in Galilei's time the failure to observe it was a strong evidence for geocentrism. $\endgroup$
    – vsz
    May 29, 2023 at 17:02
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    $\begingroup$ @vsz: I like a theory I read about, put forth by IIRC Tycho Brahe, which suggested that the planets revolve around the Sun, but the stars revolve around the Earth. This model fit observations just as well as the heliocentric model, and didn't require blind belief in interstellar distnaces that are many orders of magnitude larger than any distances between objects in the Solar System, nor blind belief in stars that were many orders of magnitude larger thant the Sun. $\endgroup$
    – supercat
    May 29, 2023 at 17:32

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