Ptolemy to the Rescue?

In order to preserve the geocentric cosmology of the time and to account for retrograde motion of Mars, Ptolemy had to make a model of planetary motion that invoked the use of epicycles. An epicycle is basically a little "wheel" that orbits on a bigger wheel.
The use of epicycles as a desperate attempt to preserve geocentric cosmology makes the orbits of planets very complicated and violates the scientific search for simplicity.
Visualization of epicycles:

Epicycles and Planetary Motion

In the (nutso) model of Ptolemy, planets were attached, not to the concentric spheres themselves, but to circles attached to the concentric spheres, as illustrated in the adjacent diagram. These circles were called "Epicycles", and the concentric spheres to which they were attached were termed the "Deferents". Then, the centers of the epicycles executed uniform circular motion as they went around the deferent at uniform angular velocity, and at the same time the epicyles (to which the planets were attached) executed their own uniform circular motion.

The net effect was as illustrated in the following animation. As the center of the epicycle moves around the deferent at constant angular velocity, the planet moves around the epicycle, also at constant angular velocity. The apparent position of the planet on the celestial sphere at each time is indicated by the line drawn from the earth through the planet and projected onto the celestial sphere. The resulting apparent path against the background stars is indicated by the blue line.

Now, in this tortured model one sees that it is possible to have retrograde motion and varying brightness, since at times as viewed from the earth the planet can appear to move "backward" on the celestial sphere. Obviously, the distance of the planet from the Earth also varies with time, which leads to variations in brightness. Thus, the idea of uniform circular motion is saved (at least in some sense) by this scheme, and it allows a description of retrograde motion and varying planetary brightness.

The main problem, is that a single epicycle does not really quantitatively account for the observedc 83 days of retrograde motion, every 786 days for Mars. Therefore ...

More Sophisticated Epicycles: The Ptolemaic Universe

However, in practice, even this was not enough to account for the detailed motion of the planets on the celestial sphere! In more sophisticated epicycle models further "refinements" were introduced:

In some cases, epicycles were themselves placed on epicycles, as illustrated in the adjacent figure.
In actual models, the center of the epicycle moved with uniform circular motion, not around the center of the deferent, but around a point that was displaced by some distance from the center of the deferent.

That ancient astronomers could convince themselves that this elaborate scheme still corresponded to "uniform circular motion" is testament to the power of three ideas that we now know to be completely wrong , but that were so ingrained into the known cultural truth of this age that they were essentially never questioned:

All motion in the heavens is uniform circular motion.
The objects in the heavens are made from perfect material, and cannot change their intrinsic properties (e.g., their brightness).
The Earth is at the center of the Universe.

These ideas concerning uniform circular motion and epicycles were catalogued by Ptolemy in 150 A.D. His book was called the "Almagest" (literally, "The Greatest"), and this picture of the structure of the Solar System has come to be called the "Ptolemaic Universe".

The Ptolemaic Moment will stand for about 1500 years with this model for planetary orbits remains unchallenged, partly because Ptolemy's model did improve the precision to which planetary positions could be predicted. Remember, however, the precision at measuring planetary positions was quite poor and therefore there was a large random error involved when comparing predictions with observations.

The Ptolomaic model represents an excellent example of "cultural" bias in science. Most citizens of the cultural wanted to accept the idea of the Earth being the center of the Universe (i.e. the earth was special) and therefore observations were molded in such a way so as to preserve that cultural norm. (Things may not be very different today).

Recall, there was a competing model of the time, that of Aristarchus of Samos (280 BC):

( Aristarchus of Samos (c. 310-230 BC), Greek astronomer, first to maintain that the Earth rotates and revolves around the Sun. On this ground, Cleanthes the Stoic declared that he ought to be indicted for impiety.

Aristarchus' advanced ideas on the movement of the Earth are known from Archimedes and Plutarch; his only extant work is a short treatise, "On the Sizes and Distances of the Sun and Moon." The values he obtained, by using geometry, are inaccurate, because of faulty observations. Aristarchus found a more precise value for the length of the solar year.)

Aristarchus was extremely clever and also figured out a way to determine the ratio of distances between the earth and the moon and the earth and the sun. While wrong, at least he reasoned that the sun was much farther from the earth than the moon.