The eighth sphere had no deferential or epicycle, but the fixed stars were attached to it. In the hipparchic system, the epicycle rotated and rotated with a regular movement along the deferent. Ptolemy, however, found that he could not reconcile this with the Babylonian observational data he had; In particular, the shape and size of the apparent retrogrades differed. The angular velocity at which the epicycle moved was not constant unless he measured it from another point, which he called the equant. It was the angular velocity at which the deferent moved around the point halfway between the equent and the Earth (the eccentric) that was constant; The center of the epicycle swept equal angles over equal times only outside the equant. It was the use of equanta to decouple the uniform motion of the center from circular deference that characterized the Ptolemaic system. If his values for the deferent rays with respect to the Earth-Sun distance had been more accurate, the epicycle sizes would all have approximated the Earth-Sun distance. Although all the planets were considered separately, they were all connected in a particular way: the lines drawn by the body through the epicentric center of all the planets were all parallel, as well as the line drawn from the sun to the Earth, along which Mercury and Venus were located. This means that all bodies rotate in their epicycles in parallel with Ptolemy`s sun (i.e.
they are all exactly one year old). [ref. needed]. Circular orbits, called epicycles, were formed in circular orbits. This astronomical system culminated in the Almagest of Ptolemy, who worked in Alexandria in the 2nd century AD. Copernican innovation simplified the system somewhat, but Copernicus` astronomical tables were still based on circular orbits and epicycles. Kepler. In the hipparchic and Ptolemaic systems, planets are thought to move in a small circle called an epicycle, which in turn moves along a larger circle called a deferens. The two circles rotate clockwise and are roughly parallel to the plane of the Sun`s apparent orbit beneath these systems (ecliptic).
Despite the fact that the system is considered geocentric, none of the circles were exactly concentric with Earth. Specifically, the motion of each planet was centered on a planet-specific point, slightly away from Earth, called eccentric. The orbits of the planets in this system are similar to those of the epitrochoids. Apollonius introduced an alternative “epicyclic” model in which the planet revolves around a point that itself orbits in a circle (the “deferent”) centered on or near the Earth. As Apollonius knew, his epicyclic model is geometrically equivalent to an eccentric. These models were well suited to other phenomena. The ancients worked from a geocentric point of view for the simple reason that the earth was where they stood and looked at the sky, and it is the sky that seems to move while the ground seems motionless and stable under your feet. Some Greek astronomers (e.g. Aristarchus of Samos) hypothesized that planets (including Earth) orbited the sun, but the optics (and specific mathematics – Isaac Newton`s law of gravity, for example) needed to provide data that would convincingly support the heliocentric model did not exist in Ptolemy`s time and would not appear until more than fifteen hundred years after his time. Moreover, Aristotelian physics was not designed with this type of calculation in mind, and Aristotle`s philosophy concerning the sky was completely at odds with the concept of heliocentrism.
It wasn`t until Galileo observed Jupiter`s moons on January 7, 1610, and the phases of Venus in September 1610, that the heliocentric model received wide support among astronomers, who also accepted the idea that planets are individual worlds orbiting the sun (i.e. the earth is also a planet). Johannes Kepler formulated his Three Laws of Planetary Motion, which describe the orbits of the planets in our solar system with remarkable accuracy, using a system that uses elliptical rather than circular orbits. Kepler`s three laws are still taught in physics and astronomy classes today, and the wording of these laws has not changed since Kepler first formulated them four hundred years ago. The ability of Newtonian mechanics to solve problems in orbital mechanics is illustrated by the discovery of Neptune. Analysis of the observed perturbations in the orbit of Uranus provided estimates of the suspicious planet`s position to a degree to which it was found. This could not have been achieved with deferential/epicyclic methods. Nevertheless, Newton published the theory of lunar motion in 1702, which used an epicycle and remained in use in China until the nineteenth century. Later tables based on Newton`s theory could have approximated accuracy in the arc minute range.
 . rotating and off-center circles called eccentrics; epicycles, small circles whose centres move uniformly around the circumference of circles of greater radius (deferents); and Ecuador. However, the equant broke with the main hypothesis of ancient astronomy because it separated the state of uniform motion from that of constant distance from the center. To correct these irregularities, Copernicus introduced epicycle after epicycle in lunar orbit. Claudius Ptolemy refined the concept of reverence and introduced the equant as a mechanism that explains the speed fluctuations in the motions of the planets. The empirical methodology he developed proved to be extraordinarily accurate for his time and was still in use in the time of Copernicus and Kepler. It is important to clarify that a heliocentric model as a system for tracking and predicting the motions of celestial bodies is not necessarily more accurate than a geocentric model when considered strictly circular orbits. A heliocentric system would require more complicated systems to compensate for the displacement of the reference point.
It was only with Kepler`s proposal of elliptical orbits that such a system became more accurate than a simple epicyclic geocentric model.  . based on eccentric circles and epicycles. (An eccentric circle is a circle that lies slightly off-center from the Earth, and an epicycle is a circle that is carried over another circle and moves.) This innovation is generally attributed to Apollonius of Perga (c. 220 BC). A.D.) attributed, but is inconclusive. The epicycles worked very well and were very accurate because, as Fourier`s analysis later showed, any smooth curve can be approached arbitrarily with a sufficient number of epicycles. However, they fell out of favor with the discovery that planetary motions were largely elliptical from a heliocentric frame of reference, leading to the discovery that gravity, obeying a simple inverse-square law, can better explain all planetary motions. As a system used for the most part to justify the geocentric model, with the exception of Copernicus` cosmos, the deferential and epicycle models were preferred to the heliocentric ideas proposed by Kepler and Galileo. The Church approved of this model because it promoted its central dogma.
 Later adopters of the epicyclical model, such as Tycho Brahe, who took into account the writings of the Church in the creation of his model, were viewed even more positively. The Tychonic model was a hybrid model that mixed geocentric and heliocentric properties with a motionless Earth with the Sun and Moon and planets orbiting the Sun. For Brahe, the idea of a rotating and moving earth was impossible, and the scriptures always had to come first and be respected.  When Galileo tried to challenge Tycho Brahe`s system, the Church was not satisfied with his views. Galileo`s publication did not help his case in his trial. The apparent movement of celestial bodies with respect to time is cyclical in nature. Apollonius of Perga realized that this cyclic variation could be visually represented by small circular orbits or epicycles rotating in larger circular orbits or deferents. Hipparchus calculated the required orbits. In ancient models, deferents and epicycles did not represent orbits in the modern sense, but a complex series of circular orbits whose centers are separated by a certain distance in order to approximate the observed motion of celestial bodies. In the hipparchic, Ptolemaic and Copernican systems of astronomy, the epicycle (from the Ancient Greek ἐπίκυκλος (epíkuklos) “on the circle”, meaning “circle moving on another circle”) was a geometric model used to explain variations in speed and direction of the apparent motion of the moon, sun, and planets.
In particular, it explained the apparent backward motion of the five planets known at the time. Secondarily, he also explained the changes in the apparent distances of Earth`s planets. rotated around small circles called epicycles at a uniform speed, while the center of the epicyclic circle revolved around the Earth in a large circle called deferens. Other variations in motion have been explained by moving the centers of the deferent for each planet on Earth a short distance. Another problem is that the models themselves advised against crafting. In a deferential and epicyclic model, the parts of the whole are connected to each other. Changing a setting to improve the fit in one place would disrupt the fit in another.