How did ptolemy attempt to explain the apparent retrograde or backwards motion of several of the planets?

In the Ptolemaic system, each planet rotates evenly along a circular path (epicycle) whose center rotates along a larger circular path (deferent) around the Earth. Since one half of an epicycle runs counter to the general movement of the deviating path, the combined movement sometimes seems to slow down or even reverse (backward) direction. While these epicycles didn’t explain the motion of the planets perfectly, it was the most accurate model until Kepler’s laws simplified things. Another version of the model, suitable for the moon, gradually shifted the direction of the line from apogee to perigee.

He argued that planets move on two circles, a deferent and an epicycle.

How did the Ptolemaic model explain the planet’s declining motion?

Ptolemy accepted Aristotle’s idea that the sun and planets revolve around a spherical earth, a geocentric view. It was difficult for ancient astronomers to explain because they believed that the Earth was the center of the universe. The planets orbit the sun at different speeds, retrograde motion is natural and occurs when the Earth seems to overtake or pass another planet in its orbit. Apparent backward movement is the apparent movement of a planet over a period of a few weeks to a few months when it appears to be moving west relative to the stars in the sky.

The heliocentric model explains the backward movement, as Mars only seems to move backwards when the Earth passes it in its orbit around the Sun.

How did Ptolemy try to explain the apparent downward or declining movement of several planets?

The “natural expectation for ancient societies was that the celestial bodies (sun, moon, planets, and stars) must move smoothly along the “perfect path” possible, a circle. The geocentric model uses a system of epicycles to explain the backward motion, with the planets moving around small orbits, which in turn moved around larger circular orbits around the Earth. In the 1500s, Copernicus explained the declining movement with a much simpler, heliocentric theory that was largely correct.