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I understand that both Copernicus and Ptolemy needed epicycles, but one might assume that a heliocentric system would need much smaller corrections to account for the non-circular motion of the planets. This is however not mentioned as an important reason to favor Copernicus’ system. Is this not the case?

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I gather the worse problem for Copernicus was Mercury. Not only is Mercury's orbit quite elliptical, it has relatively high precession due to the gravitational influence of other planets.

I found a quote attributes to Copernicus in this document (sorry, only direct link available) where he says :

It is interesting to note that Copernicus had difficulty explaining Mercury’s motion and once commented that ”this planet has . .. influenced many perplexities and labours on us in our investigation of its wanderings".

This is apparently originally from this source :

  • Baum, Richard and Sheehan, William.In Search of Planet Vulcan: The Ghost in Newton’s Clockwork Universe. Plenum Trade, New York. 1997.

The gravitational perturbation of the planets, particularly by Jupiter, and the concept of barycenter for even ideal elliptical orbits required the work of Newton (following Kepler's concept of elliptical orbits) to explain. Up until Newton there was no solid theory for the motion at all and people were "just" (poor choice of words) trying to find the simplest model. Copernicus provided a simple model, but also a less accurate one.

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It all depends what you mean by “corrections.” Indeed, Copernicus’s model is not geocentric, which should mean that it’s more precise, except that it wasn’t. His model basically is just an adaptation of Ptolemy’s model, but “translated” (in the geometrical sense) to have the Sun near the centre, and with slightly different parametres.

Because that’s another misconception : Copernicus’s model is not heliocentric but rather heliostatic, which means that the Sun is not in the centre, but simply fixed, and near the centre of planetary orbits. The distance between the Sun and the centre of a planet’s orbit is its eccentricity, and it is different for each planet.

To top it all off, Copernicus introduced additional epicycles, so his model is actually more complicated than Ptolemy’s. For example, Saturn moves on a small epicycle of radius $r’ = 285$ (arbitrary) units, which itself moves on the deferent of radius $R = 10,000$ units (the same size in arbitrary units for all planets). This deferent is not centred on the Sun, but $e = 854$ units from it. Finally, the Earth is $r = 1,090$ units from the Sun. One then has four line segments where one starts where the other one ends, and the goal is to find the angle between the two ends, knowing all the other angles. With Jupiter, terms change to $r’ = 229$ units, $e = 687$ units, and $r = 1916$ units.

The diagram below shows how it would look like from the northern pole of the ecliptic. All movements are counterclockwise. The two circular blue arrows mean that the two associated movements are done at the same rate, different from one planet to another, and called its anomaly. The values are very close to those in Ptolemy’s model.

Copernicus’s planetary model

In order to calculate the precise longitude of the planet (angle ♈–Earth–planet), one had to introduce additional lines and points in this drawing. Fortunately for his readers, Copernicus—like Ptolemy before him—drew up tables giving the value of each angle from the time or angle it depended upon, so no real mathematical/trigonometric work was needed from the reader.

Copernicus’s model was slow in its adoption, because it did not predict planetary positions better than Ptolemy’s—it was actually worse, although astronomers didn’t realize it right away—and because, as I said, it was more complicated that Ptolemy’s model. Also, it did not introduce anything new: in other words, there was no up-to-then-unexplained phenomenon that Copernicus’s model explained successfully.

So, astronomers had the choice of following a simple model that gave good results (Ptolemy’s), or a more complicated model that gave basically the same results (Copernicus’s)… The choice was easy! Add to that the anonymous preface to De Revolutionibus, which basically said that “the model presented wherein is only a mathematical exercise and does not reflect reality,” and you have a recipe for disaster for the new theory.

It’s only after Galileo showed, with his telescope, the phases of Venus and the moons of Jupiter, that some people started to be convinced—though there was some scepticism about his telescope, as some people thought it was a gimmick/trick that did not show true things, but that’s a story of a different time.

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To the authors of both systems, Mars occasionally seemed to move backward in its orbit, so epicycles were needed to account for this. It was not until Newton discovered the laws of gravity that it was realised that Mars was compelled to move slower than the Earth and therefore was sometimes overtaken by the Earth in the journey around the sun. The impression that Mars was moving backwards was of course an illusion, but still had to be accounted for. Copernicus was unaware of Newton's laws, but at least he realised that the orbits of the planets were heliocentric rather than geocentric. He also realised that Mars had further to travel than the Earth.

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