Epicycles are Real

A reader cites Carl Sagan's 1980 Cosmos a personal journey Ep. 3. Harmony of the Worlds (video, transcript).

This little machine shows Ptolemy’s model. The planets were imagined to go around the Earth, attached to perfect crystal spheres — but not attached directly to the spheres, but indirectly, through a kind of off-center wheel. The sphere turns, the little wheel rotates, and — as seen from the Earth — Mars does its loop-the-loop. This model permitted reasonably accurate predictions of planetary motion: where a planet would be on a given day. Certainly good enough predictions for the precision of measurement in Ptolemy’s time and much later. Supported by the church through the Dark Ages, Ptolemy’s model effectively prevented the advance of astronomy for 1,500 years.

Finally, in 1543, a quite different explanation of the apparent motion of the planets was published by a Polish cleric named Nicolaus Copernicus. Its most daring feature was the proposition that the sun, not the Earth, was the center of the universe. The Earth was demoted to just one of the planets. The retrograde (or loop-the-loop) motion happens as the Earth overtakes Mars in its orbit. You can see that, from the standpoint of the Earth, Mars is now going slightly backwards and now it is going in its original direction. This Copernican model worked at least as well as Ptolemy’s crystal spheres, but it annoyed an awful lot of people. The Catholic Church later put Copernicus’s work on its list of forbidden books.

So Ptolemy developed a reasonably accurate model, but it was somehow used by the Catholic Church to keep everyone in the Dark Ages?

No, this is mostly nonsense. The Church only objected to nine sentences in the Copernicus book, and not to publication of the model. Ptolemy did not require crystal spheres.

The Wikipedia article on epicycles is much more sensible:

Epicycles worked very well and were highly accurate, because, as Fourier analysis later showed, any smooth curve can be approximated to arbitrary accuracy with a sufficient number of epicycles. However, they fell out of favor with the discovery that planetary motions were largely elliptical from a heliocentric frame of reference, which led to the discovery that gravity obeying a simple inverse square law could better explain all planetary motions.

The first Ptolemy epicycle for each planet is the orbit of the Earth, and is used to explain the retrograde modtion. People can argue that this epicycle is not real, because Mars never really goes backwards. But Mars does go backwards, as viewed from the Earth, and that is what Ptolemy was modeling.

Subsequent epicycles were used to model variations in speed and direction from uniform circular motion. The Copernicus model did not need the first epicycle, as the Earth had its own orbit, but needed the subseqent epicycles, as deviations from uniform circular motion was observable at the time.

As Sagan later explained, Tycho Brahe made the observations that enabled models superior to Ptolemy's. Astronomy was not held back by epicycles, but by a lack of more precise data. That data came from Tycho, and later the invention of the telescope

Sagan pushes his anti-religion beliefs, but all these advances took places in Christendom, and not elsewhere.