Deferent and epicycle
In the Hipparchian, Ptolemaic, and Copernican systems of astronomy, the epicycle was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets. In particular it explained the apparent retrograde motion of the five planets known at the time. Secondarily, it also explained changes in the apparent distances of the planets from the Earth.
The model is often attributed to Apollonius of Perga, who was active at the end of the 3rd century BC. It was developed by Apollonius of Perga and Hipparchus of Rhodes, who used it extensively, during the 2nd century BC, then formalized and extensively used by Ptolemy in his 2nd century AD astronomical treatise the Almagest.
Epicyclical motion is used in the Antikythera mechanism, an ancient Greek astronomical device, for compensating for the elliptical orbit of the Moon, moving faster at perigee and slower at apogee than circular orbits would, using four gears, two of them engaged in an eccentric way that quite closely approximates Kepler's second law.
Epicycles worked very well and were highly accurate, because, as Fourier analysis later showed, any smooth curve can be approximated to arbitrary accuracy 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, which led to the discovery that gravity obeying a simple inverse square law could better explain all planetary motions.
Introduction
In both Hipparchian and Ptolemaic systems, the planets are assumed to move in a small circle called an epicycle, which in turn moves along a larger circle called a deferent. Both circles rotate eastward and are roughly parallel to the plane of the Sun's apparent orbit under those systems. Despite the fact that the system is considered geocentric, neither of the circles were centered on the earth, rather each planet's motion was centered at a planet-specific point slightly away from the Earth called the eccentric. The orbits of planets in this system are similar to epitrochoids, but are not exactly epitrochoids because the angle of the epicycle is not a linear function of the angle of the deferent.In the Hipparchian system the epicycle rotated and revolved along the deferent with uniform motion. However, Ptolemy found that he could not reconcile that with the Babylonian observational data available to him; in particular, the shape and size of the apparent retrogrades differed. The angular rate at which the epicycle traveled was not constant unless he measured it from another point which is now called the equant. It was the angular rate at which the deferent moved around the point midway between the equant and the Earth that was constant; the epicycle center swept out equal angles over equal times only when viewed from the equant. It was the use of equants to decouple uniform motion from the center of the circular deferents that distinguished the Ptolemaic system. For the outer planets, the angle between the center of the epicycle and the planet was the same as the angle between the Earth and the Sun.
Ptolemy did not predict the relative sizes of the planetary deferents in the Almagest. All of his calculations were done with respect to a normalized deferent, considering a single case at a time. This is not to say that he believed the planets were all equidistant, but he had no basis on which to measure distances, except for the Moon. He generally ordered the planets outward from the Earth based on their orbit periods. Later he calculated their distances in the Planetary Hypotheses and summarized them in the first column of this table:
Had his values for deferent radii relative to the Earth–Sun distance been more accurate, the epicycle sizes would have all approached the Earth–Sun distance. Although all the planets are considered separately, in one peculiar way they were all linked: the lines drawn from the body through the epicentric center of all the planets were all parallel, along with the line drawn from the Sun to the Earth along which Mercury and Venus were situated. That means that all the bodies revolve in their epicycles in lockstep with Ptolemy's Sun.
Babylonian observations showed that for superior planets the planet would typically move through in the night sky slower than the stars. Each night the planet appeared to lag a little behind the stars, in what is called prograde motion. Near opposition, the planet would appear to reverse and move through the night sky faster than the stars for a time in retrograde motion before reversing again and resuming prograde. Epicyclic theory, in part, sought to explain this behavior.
The inferior planets were always observed to be near the Sun, appearing only shortly before sunrise or shortly after sunset. Their apparent retrograde motion occurs during the transition between evening star into morning star, as they pass between the Earth and the Sun.
History
When ancient astronomers viewed the sky, they saw the Sun, Moon, and stars moving overhead in a regular fashion. Babylonians did celestial observations, mainly of the Sun and Moon as a means of recalibrating and preserving timekeeping for religious ceremonies. Other early civilizations such as the Greeks had thinkers like Thales of Miletus, the first to document and predict a solar eclipse, or Heraclides Ponticus. They also saw the "wanderers" or "planetai". The regularity in the motions of the wandering bodies suggested that their positions might be predictable.The most obvious approach to the problem of predicting the motions of the heavenly bodies was simply to map their positions against the star field and then to fit mathematical functions to the changing positions. The introduction of better celestial measurement instruments, such as the introduction of the gnomon by Anaximander, allowed the Greeks to have a better understanding of the passage of time, such as the number of days in a year and the length of seasons, which are indispensable for astronomic measurements.
The ancients worked from a geocentric perspective for the simple reason that the Earth was where they stood and observed the sky, and it is the sky which appears to move while the ground seems still and steady underfoot. Some Greek astronomers speculated that the planets orbited the Sun, but the optics necessary to provide data that would convincingly support the heliocentric model did not exist in Ptolemy's time and would not come around for over fifteen hundred years after his time. Furthermore, Aristotelian physics was not designed with these sorts of calculations in mind, and Aristotle's philosophy regarding the heavens was entirely at odds with the concept of heliocentrism. It was not until Galileo Galilei observed the moons of Jupiter on 7 January 1610, and the phases of Venus in September 1610, that the heliocentric model began to receive broad support among astronomers, who also came to accept the notion that the planets are individual worlds orbiting the Sun. Johannes Kepler formulated his three laws of planetary motion, which describe the orbits of the planets in the Solar System to a remarkable degree of accuracy utilizing a system that employs elliptical rather than circular orbits. Kepler's three laws are still taught today in university physics and astronomy classes, and the wording of these laws has not changed since Kepler first formulated them four hundred years ago.
The apparent motion of the heavenly bodies with respect to time is cyclical in nature. Apollonius of Perga realized that this cyclical variation could be represented visually by small circular orbits, or epicycles, revolving on larger circular orbits, or deferents. Hipparchus calculated the required orbits. Deferents and epicycles in the ancient models did not represent orbits in the modern sense, but rather a complex set of circular paths whose centers are separated by a specific distance in order to approximate the observed movement of the celestial bodies.
Claudius Ptolemy refined the deferent-and-epicycle concept and introduced the equant as a mechanism that accounts for velocity variations in the motions of the planets. The empirical methodology he developed proved to be extraordinarily accurate for its day and was still in use at the time of Copernicus and Kepler. A heliocentric model is not necessarily more accurate as a system to track and predict the movements of celestial bodies than a geocentric one when considering strictly circular orbits. A heliocentric system would require more intricate systems to compensate for the shift in reference point. It was not until Kepler's proposal of elliptical orbits that such a system became increasingly more accurate than a mere epicyclical geocentric model.
Owen Gingerich describes a planetary conjunction that occurred in 1504 and was apparently observed by Copernicus. In notes bound with his copy of the Alfonsine Tables, Copernicus commented that "Mars surpasses the numbers by more than two degrees. Saturn is surpassed by the numbers by one and a half degrees." Using modern computer programs, Gingerich discovered that, at the time of the conjunction, Saturn indeed lagged behind the tables by a degree and a half and Mars led the predictions by nearly two degrees. Moreover, he found that Ptolemy's predictions for Jupiter at the same time were quite accurate. Copernicus and his contemporaries were therefore using Ptolemy's methods and finding them trustworthy well over a thousand years after Ptolemy's original work was published.
When Copernicus transformed Earth-based observations to heliocentric coordinates, he was confronted with an entirely new problem. The Sun-centered positions displayed a cyclical motion with respect to time but without retrograde loops in the case of the outer planets. In principle, the heliocentric motion was simpler but with new subtleties due to the yet-to-be-discovered elliptical shape of the orbits. Another complication was caused by a problem that Copernicus never solved: correctly accounting for the motion of the Earth in the coordinate transformation. In keeping with past practice, Copernicus used the deferent/epicycle model in his theory but his epicycles were small and were called "epicyclets".
In the Ptolemaic system the models for each of the planets were different, and so it was with Copernicus' initial models. As he worked through the mathematics, however, Copernicus discovered that his models could be combined in a unified system. Furthermore, if they were scaled so that the Earth's orbit was the same in all of them, the ordering of the planets we recognize today easily followed from the math. Mercury orbited closest to the Sun and the rest of the planets fell into place in order outward, arranged in distance by their periods of revolution.
Although Copernicus' models reduced the magnitude of the epicycles considerably, whether they were simpler than Ptolemy's is moot. Copernicus eliminated Ptolemy's somewhat-maligned equant but at a cost of additional epicycles. Various 16th-century books based on Ptolemy and Copernicus use about equal numbers of epicycles. The idea that Copernicus used only 34 circles in his system comes from his own statement in a preliminary unpublished sketch called the Commentariolus. By the time he published De revolutionibus orbium coelestium, he had added more circles. Counting the total number is difficult, but estimates are that he created a system just as complicated, or even more so. Koestler, in his history of man's vision of the universe, equates the number of epicycles used by Copernicus at 48. The popular total of about 80 circles for the Ptolemaic system seems to have appeared in 1898. It may have been inspired by the non-Ptolemaic system of Girolamo Fracastoro, who used either 77 or 79 orbs in his system inspired by Eudoxus of Cnidus. Copernicus in his works exaggerated the number of epicycles used in the Ptolemaic system; although original counts ranged to 80 circles, by Copernicus's time the Ptolemaic system had been updated by Peurbach toward the similar number of 40; hence Copernicus effectively replaced the problem of retrograde with further epicycles.
Copernicus' theory was at least as accurate as Ptolemy's but never achieved the stature and recognition of Ptolemy's theory. What was needed was Kepler's elliptical-orbit theory, not published until 1609 and 1619. Copernicus' work provided explanations for phenomena like retrograde motion, but really did not prove that the planets actually orbited the Sun.
Ptolemy's and Copernicus' theories proved the durability and adaptability of the deferent/epicycle device for representing planetary motion. The deferent/epicycle models worked as well as they did because of the extraordinary orbital stability of the solar system. Either theory could be used today had Gottfried Wilhelm Leibniz and Isaac Newton not invented calculus.
According to Maimonides, the now-lost astronomical system of Ibn Bajjah in 12th century Andalusian Spain lacked epicycles. Gersonides of 14th century France also eliminated epicycles, arguing that they did not align with his observations. Despite these alternative models, epicycles were not eliminated until the 17th century, when Johannes Kepler's model of elliptical orbits gradually replaced Copernicus' model based on perfect circles.
Newtonian or classical mechanics eliminated the need for deferent/epicycle methods altogether and produced more accurate theories. By treating the Sun and planets as point masses and using Newton's law of universal gravitation, equations of motion were derived that could be solved by various means to compute predictions of planetary orbital velocities and positions. If approximated as simple two-body problems, for example, they could be solved analytically, while the more realistic n-body problem required numerical methods for solution.
The power of Newtonian mechanics to solve problems in orbital mechanics is illustrated by the discovery of Neptune. Analysis of observed perturbations in the orbit of Uranus produced estimates of the suspected planet's position within a degree of where it was found. This could not have been accomplished with deferent/epicycle methods. Still, Newton in 1702 published Theory of the Moon's Motion which employed an epicycle and remained in use in China into the nineteenth century. Subsequent tables based on Newton's Theory could have approached arcminute accuracy.