Ptolemaic system

In astronomy, the geocentric model (also known as geocentrism, or thePtolemaic system) is a description of the universe where earth is at the orbital center of all the celestial bodies. This model served as the predominant cosmological system in many ancient civilizations such as ancient Greece including the noteworthy systems of Aristotle (see Aristotelian physics) and Ptolemy. As such, they believed that the Sun, Moon, stars, and naked eye planets circled Earth.

Ptolemaic system

Pages from 1550 Annotazione on Sacrobosco's Tractatus de Sphaera, showing the Ptolemaic system.

In the Ptolemaic system, each planet is moved by a system of two spheres: one called its deferent, the other, its epicycle. The deferent is a circle whose center point, called the eccentric and marked in the diagram with an X, is removed from the Earth. The original purpose of the eccentric was to account for the differences of the lengths of the seasons (autumn is the shortest by a week or so), by placing the Earth away from the center of rotation of the rest of the universe. Another sphere, the epicycle, is embedded inside the deferent sphere and is represented by the smaller dotted line to the right. A given planet then moves around the epicycle at the same time the epicycle moves along the path marked by the deferent. These combined movements cause the given planet to move closer to and further away from the Earth at different points in its orbit, and explained the observation that planets slowed down, stopped, and moved backward in retrograde motion, and then again reversed to resume normal, or prograde, motion.

The deferent-and-epicycle model had been used by Greek astronomers for centuries along with the idea of the eccentric (a deferent which is slightly off-center from the Earth), which was even older. In the illustration, the center of the deferent is not the Earth but the spot marked X, making it eccentric (from the Greek ἐκ ec- meaning "from," and κέντρον kentron meaning "center"), from which the spot takes its name. Unfortunately, the system that was available in Ptolemy's time did not quite matchobservations, even though it was considerably improved over Hipparchus' system. Most noticeably the size of a planet's retrograde loop (especially that of Mars) would be smaller, and sometimes larger, than expected, resulting in positional errors of as much as 30 degrees. To alleviate the problem, Ptolemy developed the 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 uniform speed; all other locations would see non-uniform speed, like on the Earth. By using an equant, Ptolemy claimed to keep motion which was uniform and circular, although it departed from the Platonic ideal of uniform circular motion. The resultant system, which eventually came to be widely accepted in the west, seems unwieldy to modern astronomers; each planet required an epicycle revolving on a deferent, offset by an equant which was different for each planet. It predicted various celestial motions, including the beginning and end of retrograde motion, to within a maximum error of 10 degrees, considerably better than without the equant.

The model with epicycles is in fact a very good model of an elliptical orbit with low eccentricity. The well known ellipse shape does not appear to a noticeable extent when the eccentricity is less than 5%, but the offset distance of the 'center' (in fact the focus occupied by the sun) is very noticeable even with low eccentricities as possessed by the planets.

To summarize, Ptolemy devised a system that was compatible with Aristotelian philosophy and managed to track actual observations and predict future movement mostly to within the limits of the next 1000 years of observations. The observed motions and his mechanisms for explaining them include:

The Ptolemaic System

Object(s)	Observation	Modeling mechanism
Stars	Motion of entire sky E to W in ~24 hrs ("first motion")	Stars: Daily motion E to W of sphere of stars, carrying all other spheres with it. Normally ignored; other spheres have additional motions.
Sun	Motion yearly W to E along ecliptic	Motion of sun's sphere W to E in year
Sun	Non-uniform rate along ecliptic (uneven seasons)	Eccentric orbit (sun's deferent center off Earth)
Moon	Monthly motion W to E compared to stars	Monthly W to E motion of moon's sphere
The 5 planets	General motion W to E through zodiac	Motion of deferents W to E; period set by observation of planet going around the ecliptic.
Planets	Retrograde motion	Motion of epicycle in same direction as deferent. Period of epicycle is time between retrograde motions (synodic period).
Planets	Variations in speed thru zodiac	Eccentric per planet
Planets	Variations in retrograde timing	Equants per planet (Copernicus used a pair of epicycles instead)
Planets	Size of deferents, epicycles	Only ratio between radius of deferent and associated epicycle determined; absolute distances not determined in theory.
Interior planets	Average greatest elongations of 23 (Mercury)and 46 degrees (Venus)	Size of epicycles set by these angles, proportional to distances.
Interior planets	Limited to movement near the sun	Center their deferent centers along the Sun-Earth line.
Exterior planets	Retrograde only at opposition, when brightest	Radii of epicycles aligned to Sun-Earth line

The geocentric model was eventually replaced by the heliocentric model. The earliest heliocentric model, Copernican heliocentrism, could remove Ptolemy's epicycles because the retrograde motion could be seen to be the result of the combination of Earth and planet movement and speeds. Copernicus felt strongly that equants were a violation of Aristotelian purity, and proved that replacement of the equant with a pair of new epicycles was entirely equivalent. Astronomers often continued using the equants instead of the epicycles because the former was easier to calculate, and gave the same result.

It has been determined, in fact, that the Copernican, Ptolemaic and even the Tychonic models provided identical results to identical inputs. They are computationally equivalent. It wasn't until Kepler demonstrated a physical observation that could show that the physical sun is directly involved in determining an orbit that a new model was required.
During the middle ages the Catholic Church incorporated the Ptolemaic system into the Chritian viewof the world, asa part of its general program of combining the wisdom and learning of the ancients with thpe christian religion.On this view god made the world to be at the center of everything. and to be the master of this world he had created man man in his own image . In the heavens he had established his paradise , as being thee realm to which the souls og human beings would go after their bodies died.