How did Ptolemy use epicycles to explain the changing motions of planets in the sky?

In the Ptolemaic system, Ptolemy addressed the observed irregularities in planetary motion—such as variations in speed and the phenomenon of retrograde motion—by using a combination of circular paths.12

To explain these motions while maintaining the assumption of a stationary Earth at the center of the universe, Ptolemy utilized two primary components:

• Epicycle: Each planet revolved uniformly along a smaller circular path known as an epicycle.12
• Deferent: The center of this epicycle simultaneously revolved around Earth along a larger circular path called a deferent.12

Because one half of the epicycle’s rotation moved in the opposite direction of the deferent’s path, the planet would appear from Earth to slow down, stop, or even reverse its direction (retrograde). By carefully coordinating the speeds and radii of these circles, Ptolemy created a mathematical model capable of predicting planetary positions with significant accuracy for his time. He further refined these motions by introducing the equant, a point from which the epicycle's center appeared to move at a uniform angular speed.12