The epicycles of Ptolemy are still with us.
Ptolemy's earth-centered epicyclic system was discarded centuries ago. But it still provides a handy way to visualize the motions of our planets. You just need to relabel a few of the objects. OK, you also need to gloss over some details, such as the fact that Sol's planets and their moons move in slightly different planes.
Nevertheless, it works well for illustration. Here's an example derived (poorly) from an image in Carl Sagan's ["Cosmos"](https://www.amazon.com/Cosmos-Carl-Sagan-ebook/dp/B004W0HZN4/ref=sr_1_1_twi_kin_1?s=books&ie=UTF8&qid=1514992182&sr=1-1&keywords=cosmos):

A wee epicyclic moon

If I understand correctly, one reason for discarding the Ptolemaic model was that, as the accuracy of observational data increased, the number of epicycles needed to fit that data increased. Eventually, too many epicycles, too much complexity, and too much number crunching were needed to make the system work.
Copernicus' system didn't fare any better, at first. But Kepler, after many dead-ends, finally deduced that its circular orbits needed to be replaced with ellipses. The story of how that came to pass is a big one that encompasses Kepler's mysticism, Tycho Brahe's accurate, last-of-its-kind non-telescopic data, and the lucky fact that Brahe made Kepler start w. the problem of Mars.[[1]](#epicycles_ever_after_footnote_1)
## Epicycles Today
We still use epicycles. Fourier series approximate arbitrarily complex waveforms by summing up sinusoidal functions having different frequencies and amplitudes. Just like [my poorly-researched claims about the complexity of] the Ptolemaic system, Fourier series can require a lot of sinusoids in order to accurately reproduce the signals they model. Thank goodness for fast computers.
Last year I came across a beautiful animation of a Fourier series, one that used spinning radius arms to show the construction of a square wave from a series of harmonic sinusoids.
Unfortunately, I can't find it now. But I've rebuilt it, and in the process discovered that "modern" Javascript is a very pleasant language for those who would rather inherit from classes than from prototypes.
[Updated 2018-09-09 to add a lot of overtones]

Number of overtones: 0

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[1]
For a long time I thought that my first impressions of Kepler came from Arthur Koestler's ["The Watershed"](https://www.amazon.com/Arthur-Koestlers-Sleepwalkers-Watershed-Astronomy-ebook/dp/B00IM6IJWU/ref=sr_1_6?s=books&ie=UTF8&qid=1514992222&sr=1-6&keywords=the+watershed+koestler). But now I'm not so sure. I watched "Cosmos" when it aired around 1980, and it was at least six years later that I first saw "The Watershed" in a used bookstore.[[2]](#epicycles_ever_after_footnote_2) Re-reading "Cosmos" recently, I was surprised at how much of the story it told.
"The Watershed" is an appropriate title for Koestler's book. Kepler was something of a mystic, who started out trying to prove that the orbits of the known planets corresponded to a nesting of the Platonic solids. But he was also mathematically tenacious, unsatisfied with a model that didn't fit the data. He straddled the boundary between two ways of understanding the world. His life marked a divide between two watersheds.
This quote from "Cosmos" shows Kepler struggling to reconcile his analysis of empirical data with his religious understanding of the world:
> Eventually, Kepler came to feel that his fascination with the circle had been a delusion. The Earth was a planet, as Copernicus had said, and it was entirely obvious to Kepler that the Earth, wracked by wars, pestilence, famine and unhappiness, fell short of perfection. Kepler was one of the first people since antiquity to propose that the planets were material objects made of imperfect stuff like the Earth. And if planets were “imperfect,” why not their orbits as well? He tried various oval-like curves, calculated away, made some arithmetical mistakes (which caused him at first to reject the correct answer) and months later in some desperation tried the formula for an ellipse, first codified in the Alexandrian Library by Apollonius of Perga. He found that it matched Tycho’s observations beautifully: “The truth of nature, which I had rejected and chased away, returned by stealth through the back door, disguising itself to be accepted … Ah, what a foolish bird I have been!”
Sagan, Carl. Cosmos (Kindle Locations 1167-1174). Random House Publishing Group. Kindle Edition.
Both Koestler's ["The Sleepwalkers"](https://www.amazon.com/Sleepwalkers-History-Changing-Universe-Classics-ebook/dp/B06VSXFQJH/ref=sr_1_1?s=books&ie=UTF8&qid=1514992363&sr=1-1&keywords=the+sleepwalkers+koestler) and "The Watershed" (which is excerpted from it) delve more deeply than "Cosmos" into the entertaining and eccentric characters in this story.
Tyge Brahe is the prime example. As a young man he lost his nose in a sword duel following an argument over whether he or his opponent was the better mathematician. He had a fool named Jepp, a dwarf who sat at his feet under the table during meals, to whom he tossed bits of food. He himself died after drinking too much at a banquet and being too courteous to step out to relieve himself. (Or [maybe he was poisoned!](https://www.amazon.com/Heavenly-Intrigue-Johannes-GreatestScientific-Discoveries-ebook/dp/B000FCK6Z2/ref=sr_1_2?ie=UTF8&qid=1514992995&sr=8-2&keywords=Heavenly+intrigue) Or [maybe not](https://phys.org/news/2012-11-mercury-poisoning-tycho-brahe-death.html).)
"The Watershed" also depicts Kepler and his family as strange and interesting. Kepler may have been one of the earliest authors of science fiction: his ["Somnium"](https://www.amazon.com/Somnium-Dream-Johannes-Kepler-ebook/dp/B00FASY90Y/ref=sr_1_1?ie=UTF8&qid=1515040475&sr=8-1&keywords=kepler+somnium) describes a journey to the moon, and the aliens who live there. His mother, prone to dispensing home remedies of which a few may have been hallucinogenic, was nearly burned as a witch. Kepler defended her at trial, more than once, with considerable skill:
> "The accused appeared in court, accompanied, alas, by her son, Johannes Kepler, mathematician."
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[2] Used bookstores are great. The 50-cent bin at the Wright State University bookstore once turned up a copy of "The Canterbury Tales." For less than the cost of a comic book I thought it would be fun to try to learn to read Middle English. Imagine using your Little Orphan Annie Decoder Ring for the first time, and instead of a crummy commercial you get a raunchy joke!