In the Hipparchian, Ptolemaic, and Copernican systems of astronomy, the epicycle (from Ancient Greek : ἐπίκυκλος, literally upon the circle, meaning circle moving on another circle ) 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.
It was first proposed by Apollonius of Perga 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 of Thebaid 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 favour 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.
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 clockwise and are roughly parallel to the plane of the Sun's orbit (ecliptic). Despite the fact that the system is considered geocentric, each planet's motion was not centered on the Earth but at a point slightly away from the Earth called the eccentric. The orbits of planets in this system are similar to epitrochoids.
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 he called the equant . It was the angular rate at which the deferent moved around the point midway between the equant and the Earth (the eccentric) 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.
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:
|Body||Mean size |
(in Earth radii)
|Modern value |
in Earth radii)
normalized to Sun = 1)
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 (that is, they all have exactly a one-year period).[ citation needed ]
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.
When ancient astronomers viewed the sky, they saw the Sun, Moon, and stars moving overhead in a regular fashion. They also saw the "wanderers" or "planetai" (our planets). 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 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 (e.g., Aristarchus of Samos) speculated that the planets (Earth included) orbited the Sun, but the optics (and the specific mathematics – Isaac Newton's Law of Gravitation for example) 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 (that is, that the Earth is a planet and is one among several). Johannes Kepler was able to formulate his three laws of planetary motion, which described the orbits of the planets in our solar system to a remarkable degree of accuracy; 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.
Claudius Ptolemy refined the deferent-and-epicycle concept and introduced the equant as a mechanism for accounting 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.
Owen Gingerichdescribes a planetary conjunction that occurred in 1504 that 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 towards 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 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.
The first planetary model without any epicycles was that of Ibn Bajjah (Avempace) in 12th century Andalusian Spain,but epicycles were not eliminated in Europe 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. Simple two-body problems, for example, can be solved analytically. More-complex n-body problems require 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.
According to one school of thought in the history of astronomy, minor imperfections in the original Ptolemaic system were discovered through observations accumulated over time. It was mistakenly believed that more levels of epicycles (circles within circles) were added to the models to match more accurately the observed planetary motions. The multiplication of epicycles is believed to have led to a nearly unworkable system by the 16th century, and that Copernicus created his heliocentric system in order to simplify the Ptolemaic astronomy of his day, thus succeeding in drastically reducing the number of circles.
With better observations additional epicycles and eccentrics were used to represent the newly observed phenomena till in the later Middle Ages the universe became a 'Sphere/With Centric and Eccentric scribbled o'er,/Cycle and Epicycle, Orb in Orb'.— Dorothy Stimson, The Gradual Acceptance of the Copernican Theory of the Universe, 1917
As a measure of complexity, the number of circles is given as 80 for Ptolemy, versus a mere 34 for Copernicus.The highest number appeared in the Encyclopædia Britannica on Astronomy during the 1960s, in a discussion of King Alfonso X of Castile's interest in astronomy during the 13th century. (Alfonso is credited with commissioning the Alfonsine Tables.)
By this time each planet had been provided with from 40 to 60 epicycles to represent after a fashion its complex movement among the stars. Amazed at the difficulty of the project, Alfonso is credited with the remark that had he been present at the Creation he might have given excellent advice.
As it turns out, a major difficulty with this epicycles-on-epicycles theory is that historians examining books on Ptolemaic astronomy from the Middle Ages and the Renaissance have found absolutely no trace of multiple epicycles being used for each planet. The Alfonsine Tables, for instance, were apparently computed using Ptolemy's original unadorned methods.
Another problem is that the models themselves discouraged tinkering. In a deferent-and-epicycle model, the parts of the whole are interrelated. A change in a parameter to improve the fit in one place would throw off the fit somewhere else. Ptolemy's model is probably optimal in this regard. On the whole it gave good results but missed a little here and there. Experienced astronomers would have recognized these shortcomings and allowed for them.
According to the historian of science Norwood Russell Hanson:
There is no bilaterally-symmetrical, nor eccentrically-periodic curve used in any branch of astrophysics or observational astronomy which could not be smoothly plotted as the resultant motion of a point turning within a constellation of epicycles, finite in number, revolving around a fixed deferent.— Norwood Russell Hanson, "The Mathematical Power of Epicyclical Astronomy", 1960
Any path—periodic or not, closed or open—can be represented with an infinite number of epicycles.
This is because epicycles can be represented as a complex Fourier series; so, with a large number of epicycles, very complicated paths can be represented in the complex plane.
Let the complex number
where a0 and k0 are constants, i = √ is the imaginary unit, and t is time, correspond to a deferent centered on the origin of the complex plane and revolving with a radius a0 and angular velocity
where T is the period.
If z1 is the path of an epicycle, then the deferent plus epicycle is represented as the sum
This is an almost periodic function, and is a periodic function just when the ratio of the constants kj is rational. Generalizing to N epicycles yields the almost periodic function
which is periodic just when every pair of kj is rationally related. Finding the coefficients aj to represent a time-dependent path in the complex plane, z = f(t), is the goal of reproducing an orbit with deferent and epicycles, and this is a way of "saving the phenomena" (σώζειν τα φαινόμενα).
This parallel was noted by Giovanni Schiaparelli.Pertinent to the Copernican Revolution's debate about "saving the phenomena" versus offering explanations, one can understand why Thomas Aquinas, in the 13th century, wrote:
Reason may be employed in two ways to establish a point: firstly, for the purpose of furnishing sufficient proof of some principle [...]. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results, as in astronomy the theory of eccentrics and epicycles is considered as established, because thereby the sensible appearances of the heavenly movements can be explained; not, however, as if this proof were sufficient, forasmuch as some other theory might explain them.
In part, due to misunderstandings about how deferent/epicycle models worked, "adding epicycles" has come to be used as a derogatory comment in modern scientific discussion. The term might be used, for example, to describe continuing to try to adjust a theory to make its predictions match the facts. There is a generally accepted idea that extra epicycles were invented to alleviate the growing errors that the Ptolemaic system noted as measurements became more accurate, particularly for Mars. According to this notion, epicycles are regarded by some as the paradigmatic example of bad science.Part of the problem may be due to the misconception of the epicycle as an explanation of a body's motion rather than merely a description. Toomer explains as follows,
Whereas we use "hypothesis" to denote a tentative theory which is still to be verified, Ptolemy usually means by ύπόθεσις something more like "model", "system of explanation", often indeed referring to "the hypotheses which we have demonstrated".
Copernicus added an extra epicycle to his planets, but that was only in an effort to eliminate Ptolemy's equant, which he considered a philosophical break away from Aristotle's perfection of the heavens. Mathematically, the second epicycle and the equant produce the same results, and many Copernican astronomers before Kepler continued using the equant, as the math was easier.
|Wikimedia Commons has media related to Epicycle .|
In astronomy, the geocentric model is a superseded description of the Universe with Earth at the center. Under the geocentric model, the Sun, Moon, stars, and planets all orbited Earth. The geocentric model was the predominant description of the cosmos in many ancient civilizations, such as those of Aristotle in Classical Greece and Ptolemy in Roman Egypt.
In astronomy and celestial navigation, an ephemeris gives the trajectory of naturally occurring astronomical objects as well as artificial satellites in the sky, i.e., the position over time. The etymology is from Latin ephemeris 'diary' and from Greek ἐφημερίς (ephemeris) 'diary, journal'. Historically, positions were given as printed tables of values, given at regular intervals of date and time. The calculation of these tables was one of the first applications of mechanical computers. Modern ephemerides are often computed electronically, from mathematical models of the motion of astronomical objects and the Earth. However, printed ephemerides are still produced, as they are useful when computational devices are not available.
Heliocentrism is the astronomical model in which the Earth and planets revolve around the Sun at the center of the Universe. Historically, heliocentrism was opposed to geocentrism, which placed the Earth at the center. The notion that the Earth revolves around the Sun had been proposed as early as the 3rd century BC by Aristarchus of Samos, but at least in the medieval world, Aristarchus' heliocentrism attracted little attention—possibly because of the loss of scientific works of the Hellenistic period.
The Tychonic system is a model of the Universe published by Tycho Brahe in the late 16th century, which combines what he saw as the mathematical benefits of the Copernican system with the philosophical and "physical" benefits of the Ptolemaic system. The model may have been inspired by Valentin Naboth and Paul Wittich, a Silesian mathematician and astronomer. A similar model was implicit in the calculations a century earlier by Nilakantha Somayaji of the Kerala school of astronomy and mathematics.
Nicolaus Copernicus was a Renaissance-era mathematician, astronomer, and Catholic clergyman who formulated a model of the universe that placed the Sun rather than Earth at the center of the universe. In all likelihood, Copernicus developed his model independently of Aristarchus of Samos, an ancient Greek astronomer who had formulated such a model some eighteen centuries earlier.
The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus, and others. In these celestial models, the apparent motions of the fixed stars and planets are accounted for by treating them as embedded in rotating spheres made of an aetherial, transparent fifth element (quintessence), like jewels set in orbs. Since it was believed that the fixed stars did not change their positions relative to one another, it was argued that they must be on the surface of a single starry sphere.
Equant is a mathematical concept developed by Claudius Ptolemy in the 2nd century AD to account for the observed motion of the planets. The equant is used to explain the observed speed change in planetary orbit during different stages of the orbit. This planetary concept allowed Ptolemy to keep the theory of uniform circular motion alive by stating that the path of heavenly bodies was uniform around one point and circular around another point.
De revolutionibus orbium coelestium is the seminal work on the heliocentric theory of the astronomer Nicolaus Copernicus (1473–1543) of the Polish Renaissance. The book, first printed in 1543 in Nuremberg, Holy Roman Empire, offered an alternative model of the universe to Ptolemy's geocentric system, which had been widely accepted since ancient times.
ʿAbu al-Ḥasan Alāʾ al‐Dīn ʿAlī ibn Ibrāhīm al-Ansari known as Ibn al-Shatir or Ibn ash-Shatir was an Arab astronomer, mathematician and engineer. He worked as muwaqqit in the Umayyad Mosque in Damascus and constructed a sundial for its minaret in 1371/72.
The Alfonsine tables, sometimes spelled Alphonsine tables, provided data for computing the position of the Sun, Moon and planets relative to the fixed stars.
The Copernican Revolution was the paradigm shift from the Ptolemaic model of the heavens, which described the cosmos as having Earth stationary at the center of the universe, to the heliocentric model with the Sun at the center of the Solar System. This revolution consisted of two phases; the first being extremely mathematical in nature and the second phase starting in 1610 with the publication of a pamphlet by Galileo. Beginning with the publication of Nicolaus Copernicus’s De revolutionibus orbium coelestium, contributions to the “revolution” continued until finally ending with Isaac Newton’s work over a century later.
Greek astronomy is astronomy written in the Greek language in classical antiquity. Greek astronomy is understood to include the ancient Greek, Hellenistic, Greco-Roman, and Late Antiquity eras. It is not limited geographically to Greece or to ethnic Greeks, as the Greek language had become the language of scholarship throughout the Hellenistic world following the conquests of Alexander. This phase of Greek astronomy is also known as Hellenistic astronomy, while the pre-Hellenistic phase is known as Classical Greek astronomy. During the Hellenistic and Roman periods, much of the Greek and non-Greek astronomers working in the Greek tradition studied at the Musaeum and the Library of Alexandria in Ptolemaic Egypt.
The Commentariolus is Nicolaus Copernicus's brief outline of an early version of his revolutionary heliocentric theory of the universe. After further long development of his theory, Copernicus published the mature version in 1543 in his landmark work, De revolutionibus orbium coelestium.
Islamic cosmology is the cosmology of Islamic societies. It is mainly derived from the Qur'an, Hadith, Sunnah, and current Islamic as well as other pre-Islamic sources. The Qur'an itself mentions seven heavens.
An equatorium is an astronomical calculating instrument. It can be used for finding the positions of the Moon, Sun, and planets without calculation, using a geometrical model to represent the position of a given celestial body.
Copernican heliocentrism is the name given to the astronomical model developed by Nicolaus Copernicus and published in 1543. This model positioned the Sun at the center of the Universe, motionless, with Earth and the other planets orbiting around it in circular paths, modified by epicycles, and at uniform speeds. The Copernican model displaced the geocentric model of Ptolemy that had prevailed for centuries, which had placed Earth at the center of the Universe. Copernican heliocentrism is often regarded as the launching point to modern astronomy and the Scientific Revolution.
Paul Wittich was a German mathematician and astronomer whose Capellan geoheliocentric model, in which the inner planets Mercury and Venus orbit the sun but the outer planets Mars, Jupiter and Saturn orbit the Earth, may have directly inspired Tycho Brahe's more radically heliocentric geoheliocentric model in which all the 5 known primary planets orbited the Sun, which in turn orbited the stationary Earth.
The historical models of the Solar System began during prehistoric periods and is updated to this day. The models of the Solar System throughout history were first represented in the early form of cave markings and drawings, calendars and astronomical symbols. Then books and written records then became the main source of information that expressed the way the people of the time thought of the Solar System.
The Vicarious Hypothesis, or hypothesis vicaria, was a planetary hypothesis proposed by Johannes Kepler to describe the motion of Mars. The hypothesis adopted the circular orbit and equant of Ptolemy's planetary model as well as the heliocentrism of the Copernican model. Calculations using the Vicarious Hypothesis did not support a circular orbit for Mars, leading Kepler to propose elliptical orbits as one of three laws of planetary motion in Astronomia Nova.
The Wittenberg Interpretation refers to the work of astronomers and mathematicians at the University of Wittenberg in response to the heliocentric model of the solar system proposed by Nicholas Copernicus, in his 1543 book De revolutionibus orbium coelestium. The Wittenberg Interpretation fostered an acceptance of the heliocentric model and had a part in beginning the Scientific Revolution.