With epicycles, Ptolemy took care of the looping motion of the planets, but there’s something else that needs accounting for as well. Let’s watch Mars over many years, and see what the loops look like over time. As we keep track of the retrograde loops, there is also a series of blue lines at the top of the screen, which are exactly the same as the loops you see on the bottom, but without taking the latitude into consideration.
As we watch, we see that there are different regions of the ecliptic where Mars behaves differently: there is one region in which Mars moves faster (and makes smaller loops, when it loops there), and another region where it moves slower (and makes larger loops). How can this be explained?
If the motion of Mars were simply an epicycle on a deferent circle that went right around the earth, the retrogressions would all look the same, like this (leaving out the latitudes):
So, to account for the different speeds in different parts of the orbit, Ptolemy first tried moving the deferent off-center from the earth. That way, the part of the orbit near the earth would appear faster. There are eight positions to make it easier to see the orbital motion as a whole. Here, the red lines are the apparent positions, as seen by the earth, and as you can see, the angle on the left, for the motion over a couple of months, is much larger than on the right, for the same amount of time. Now, let’s watch the changes in the map of retrogressions, to see where this gets us. As you can see, it does cause the retrogressions to bunch up more in the right places, but the widths of the retrogressions are off.
So, Ptolemy made a second adjustment to the deferent: he split the center in two. There would now be two centers &emdash; a center of distances, and a center of motion. The planet, or rather, the center of the planet’s epicycle would move in the shape of a circle around the center of distances, but its speed would be determined by moving at a constant angular rate around the center of motion. This center of motion was called the “equant.” Let’s watch this transition a few times. The angles of the blue lines from the equant remain 45 degrees apart, but the planetary positions change as the white circle they are on moves, which then changes where they appear to be as the earth watches them. Now, watch what happens to the retrogressions map as we go from the off-center circle, to the equant model. This looks pretty close to the observations!
Let’s watch the transitions again, from a simple circle (a concentric), to an off-center circle (an eccentric), to the use of an equant.
With this model, a planet moving on an epicycle, on a circle that does not have the earth at the center, moving at a constant speed as perceived by yet another point (the equant), Ptolemy decided his job was done. Oh, and remember: all of that apparatus spins around the earth every single day!