Well, epicycle model was exact mathematical analogy of observed motion of planets.
From the newscientist.com
The epicycle, meanwhile, rotated and revolved along the deferent with uniform motion.
From the en.wikipedia.org
The planets were thought by Ptolemy to orbit the Earth in deferent and epicycle motions.
From the en.wikipedia.org
In fact, he did this separately for the eccentric and the epicycle model.
From the en.wikipedia.org
The deferent-and-epicycle model can mimic the eccentric model, as shown by Apollonius'theorem.
From the en.wikipedia.org
Later, the belief that all spheres are concentric was forsaken in favor of Ptolemy's deferent and epicycle.
From the en.wikipedia.org
Astronomers responded by introducing the deferent and epicycle, which caused a planet to vary its distance.
From the en.wikipedia.org
Much of the content of the epicycle system was taken over by its successors and is still in use today.
From the en.wikipedia.org
The circle acted as a deferent, carrying an epicycle along which the Moon was assumed to move in a retrograde direction.
From the en.wikipedia.org
More examples
A circle that rolls around (inside or outside) another circle; generates an epicycloid or hypocycloid
In the Ptolemaic system of astronomy, the epicycle (literally: on the circle in Greek) was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets. ...
A small circle whose centre is on the circumference of a larger circle; in Ptolemaic astronomy it was seen as the basis of revolution of the "seven planets", given a fixed central Earth; Any circle whose circumference rolls around that of another circle, thus creating a hypocycloid or epicycloid
(Epicycles) Ptolemaic cosmology includes a smaller circular planetary motion as well as a larger circular orbit around the Earth. Epicycles are used to correct for discrepancies between observed planetary positions and the theoretical positions required for a simple circular orbit.
A smaller sphere, holding the planet, embedded on the surface of a larger sphere, the deferent, which was centered at the center of the universe. As the deferent rotated the epicyle also rotated producing a new motion by the combination of the motions. ...
A construct of the geocentric model of the solar system which was necessary to explain observed planetary motions. Each planet rides on a small epicycle whose center in turn rides on a larger circle (the deferent).