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3 Problems with geocentricity
Unfortunately, the system still did not quite match observations. Sometimes the size of a planet's retrograde loop (most notably that of Mars) would be smaller, and sometimes larger. Ptolemy could not explain this even when he moved deferents off-center, for the change in loop size did not match with the change in speed. This prompted Ptolemy to come up with the idea of an equant. The equant was a point near the center of a planet's orbit which, if you were to stand there and watch, the center of the planet's epicycle would always appear to move at the same speed. Therefore, the planet actually moved at different speeds at different points in its orbit. By using an equant, Ptolemy claimed to keep motion which was uniform and circular, but a lot of people didn't like it because they didn't think it was true to Plato's dictum of "uniform, circular motion". (Actually, he could have used epicycles within epicycles to explain this phenomenon, as Copernicus later did using epicyclets).
4 Replacement with Copernican system
Ptolemy's model was finally disproved by Galileo, when, using his telescope, he discovered that Venus goes through phases, just like our moon does. Under the Ptolemaic system, however, Venus can only be either between Earth and the Sun, or on the other side of the Sun (Ptolemy placed it inside the orbit of the Sun, after Mercury, but this was completely arbitrary; he could just as easily swapped Venus and Mercury and put them on the other side, or any combination of placements of Venus and Mercury, as long as they were always colinear with Earth and Sun). If that was the case, however, it would not appear to go through all phases, as was observed. If it was between Earth and Sun, it would always appear mostly dark, since the light from the sun would be falling mainly where we can't see it. On the other hand, if it was on the far side, we would only be able to see the lit side. Galileo saw it small and full, and later large and crescent. The only (reasonable) way to explain that is by having Venus orbit the Sun.
Astronomy
