Epicycles and parametric equations

Back when folk thought the planets orbited the earth,
it was noticed that Mars seemed to -go backwards-.
A resolution (Ptolemy?) was to suggest that the planets orbited in
-epicycles-; basically the figures you get from coupled circles:
if an ant moves on a circle that is -itself- moving on a circle, its
path is what I'm talking about.

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[Indeed, this is accurate (if you consider the orbits of planets to be
 circles -- they're actually ellipses):
 from the POV of Earth, Mars does move in epicycles.]
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This "coupled circles" should -immediately- suggest that what's
happening is that both planets are -circling- some third body:
the lil' circle of the epicycle is actually the orbit of the Earth.

Presumably such considerations lead Copernicus to conclude that
everything was orbiting the Sun (that, and it explained the
observations).
