Ptolemaic system
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The Ptolemaic system was a model to explain the motions of the heavens in which the earth was the centre of the universe and all other celestial bodies rotated around it, espoused by Claudius Ptolemaeus in his work, the Almagest some time around the 2nd century, C.E., and accepted for over 1,000 years by the vast majority of Europeans to be the correct cosmological model. It may be also called the geocentric model. It was overthrown by the Copernican revolution after Galileo Galilei and Copernicus discovered that the planets orbited the sun.
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The Almagest
An Epitome of the Almagest (Epitome in Ptolemaei Almagestum) was written between 1460 and 1463 by the Austrian astronomer Georg Peurbach and his famous pupil Johannes Regiomontanus at the suggestion of Cardinal Bessarion. It gave Europeans the first sophisticated understanding of Ptolemy's astronomy, and was studied by every competent astronomer of the 16th century.
Unlike earlier systems (such as 'the stars move because that is the will of the gods', or the model of concentric spheres), the Ptolemaic model explained all phenomena in the sky, while holding to Plato's dictum which states that all motions in the heavens can be explained with uniform, circular motion, and obeying Aristotelian physics.
Geocentricity
According to the Ptolemaic model, the spherical Earth is at the center of the universe. All heavenly bodies are attached to crystal spheres which rotate around Earth. The Moon is on the innermost sphere, and touches the realm of Earth, thereby contaminating it, and causing the light and dark spots and the ability to go through phases. It is not perfect like the other heavenly bodies, which shine by their own light. The planets are actually attached to 2 spheres: one sphere which is centered on Earth (the deferent), and another sphere (the epicycle) embedded within the deferent. The epicycle rotates within the deferent, causing the planet to move closer to and farther from Earth at different points in its orbit, and even to slow down, stop, and move backward (in retrograde motion). (The earlier model based on concentric spheres explained retrograde motion, but did not explain the changes in brightness caused by the change in distance). The epicycles of Venus and Mercury are always centered on a line between Earth and the Sun (Mercury being closer to Earth), which explains why they are always near it in the sky. The order of spheres from Earth outward is: Earth, Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, Stars.
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).
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.