Editing Deferent and epicycle (section)

==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.<ref>{{cite journal |last=Olmstead |first=A. T. |date=1938 |title=Babylonian Astronomy: Historical Sketch |url=https://www.jstor.org/stable/3088090 |journal=The American Journal of Semitic Languages and Literatures |volume=55 |issue=2 |pages=113–129 |doi=10.1086/amerjsemilanglit.55.2.3088090 |jstor=3088090 |s2cid=170628425 |issn=1062-0516}}</ref> Other early civilizations such as the Greeks had thinkers like [[Thales of Miletus]], the first to document and predict a solar eclipse (585 BC),<ref>{{cite journal |last=Mosshammer |first=Alden A. |date=1981 |title=Thales' Eclipse |url=https://www.jstor.org/stable/284125 |journal=Transactions of the American Philological Association |volume=111 |pages=145–155 |doi=10.2307/284125 |jstor=284125 |issn=0360-5949}}</ref> or [[Heraclides Ponticus]]. They also saw the "wanderers" or ''"planetai"'' (our [[planet]]s). The regularity in the motions of the wandering bodies suggested that their positions might be predictable.
[[File:Cassini_apparent.jpg|thumb|The complexity to be described by the geocentric model]]
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 [[function (mathematics)|mathematical functions]] to the changing positions.<ref>For an example of the complexity of the problem, see {{cite book |isbn=978-0099476443|last1=Owen Gingerich |title='The Book Nobody Read' |date=2004 |page=50|publisher=Arrow }}</ref> The introduction of better celestial measurement instruments, such as the introduction of the [[gnomon]] by Anaximander,<ref>{{cite book |author=Diogenes Laertius |title=The lives and opinions of eminent philosophers |date=September 2013 |publisher=General Books |isbn=978-1-230-21699-7 |oclc=881385989}}</ref> 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,<ref>{{cite book |last=Pedersen |first=Olaf |title=Early physics and astronomy: a historical introduction |date=1993 |publisher=Cambridge University Press |isbn=0-521-40340-5 |edition=Rev. |location=Cambridge, UK |oclc=24173447}}</ref> 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 (e.g., [[Aristarchus of Samos]]) speculated that the planets (Earth included) orbited the Sun, but the [[optics]] (and the specific mathematics – [[Isaac Newton]]'s [[Newton's law of universal gravitation|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, too). [[Johannes Kepler]] formulated his three [[Kepler's laws of planetary motion|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 [[frequency|cyclical]] in nature. [[Apollonius of Perga]] (3rd century BC) realized that this cyclical variation could be represented visually by small circular orbits, or ''epicycles'', revolving on larger circular orbits, or ''deferents''. [[Hipparchus]] (2nd century BC) 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.<ref>{{cite book |last=Evans |first=James |url=https://www.worldcat.org/oclc/729872798 |title=The history and practice of ancient astronomy |date=1998 |publisher=Oxford University Press |isbn=978-0-19-987445-3 |location=New York |oclc=729872798}}</ref>
[[File:ThomasDiggesmap.JPG|thumb|The basic simplicity of the Copernican universe, from [[Thomas Digges]]' book]]
[[Owen Gingerich]]<ref>{{cite book |isbn=978-0099476443|last1=Owen Gingerich |title='The Book Nobody Read' |date=2004 |chapter=4|publisher=Arrow }}</ref> 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,<ref>One volume of ''De Revolutionibus'' was devoted to a description of the trigonometry used to make the transformation between geocentric and heliocentric coordinates.</ref> 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.{{dubious|date=April 2023}} 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.<ref name=Nobody>{{cite book |isbn=978-0099476443|last1=Owen Gingerich |title='The Book Nobody Read' |date=2004|publisher=Arrow }}</ref>{{rp|267}} 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.<ref name=Nobody/>{{rp|54}}

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.<ref>{{cite journal |last1=Palter |first1=Robert |year=1970 |title=Approach to the History of Astronomy |journal=Studies in the History and Philosophy of Science |volume=1 |page=94 |doi=10.1016/0039-3681(70)90001-4}}</ref><ref>[[Owen Gingerich]], "Alfonso X as a Patron of Astronomy", in ''The Eye of Heaven: Ptolemy, Copernicus, Kepler'' (New York: American Institute of Physics, 1993), p. 125.</ref><ref>Gingerich, "Crisis versus Aesthetic in the Copernican Revolution", in ''Eye of Heaven'', pp.&nbsp;193–204.</ref> 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.<ref>"The popular belief that Copernicus's heliocentric system constitutes a significant simplification of the Ptolemaic system is obviously wrong&nbsp;... [T]he Copernican models themselves require about twice as many circles as the Ptolemaic models and are far less elegant and adaptable." {{cite book |last=Neugebauer |first=Otto |url=https://books.google.com/books?id=JVhTtVA2zr8C |title=The Exact Sciences in Antiquity |publisher=[[Dover Publications]] |year=1969 |isbn=978-0-486-22332-2 |edition=2 |author-link=Otto E. Neugebauer |orig-date=1957}}, p. 204. This is an extreme estimate in favor of Ptolemy.</ref> Koestler, in his history of man's vision of the universe, equates the number of epicycles used by Copernicus at 48.<ref>{{cite book |last=Koestler |first=Arthur |title=The Sleepwalkers |publisher=Arkana, [[Penguin Books]] |year=1989 |author-link=Arthur Koestler |orig-date=1959}}, p. 195</ref> 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]].<ref>Palter, ''Approach to the History of Astronomy'', pp.&nbsp;113–114.</ref> 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.<ref>{{cite book |last=Koestler |first=Arthur |title=The Sleepwalkers |publisher=Arkana, [[Penguin Books]] |year=1989 |author-link=Arthur Koestler |orig-date=1959}}, pp. 194–195.</ref>

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.
[[File:Epicycle_solaire.jpg|thumb|The deferent (''O'') is offset from the Earth (''T''). ''P'' is the center of the epicycle of the Sun ''S''.]]
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]].<ref>A deferent/epicycle model is in fact used to compute Lunar positions needed to define modern Hindu calendars. See Nachum Dershovitz and Edward M. Reingold: ''[[Calendrical Calculations]]'', Cambridge University Press, 1997, Chapter 14. ({{ISBN|0-521-56474-3}})</ref>

According to [[Maimonides]], the now-lost astronomical system of [[Ibn Bajjah]] in 12th century [[Al-Andalus|Andalusian Spain]] lacked epicycles. [[Gersonides]] of 14th century France also eliminated epicycles, arguing that they did not align with his observations.<ref>{{cite journal |last1=Goldstein |first1=Bernard R. |year=1972 |title=Theory and Observation in Medieval Astronomy |journal=Isis |volume=63 |issue=1 |pages=39–47 [40–41] |doi=10.1086/350839 |s2cid=120700705}}</ref> 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 problem]]s, 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 [[Uranus#Orbit and rotation|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.<ref>{{cite book |last=Kollerstrom |first=Nicholas |title=Newton's Forgotten Lunar Theory |publisher=Green Lion Press |year=2000 |isbn=1-888009-08-X}}</ref>