Editing Equant (section)

==Discovery and use==
Ptolemy introduced the equant in "[[Almagest]]".<ref name=Ptolemy-Almagest/> The evidence that the equant was a required adjustment to [[Aristotelian physics]] relied on observations made by himself and a certain "Theon" (perhaps, [[Theon of Smyrna]]).<ref name=Evans/>

===Hipparchus===
In models of planetary motion that precede [[Ptolemy]], generally attributed to [[Hipparchus]], the eccentric and epicycles were already a feature. The Roman writer [[Pliny the Elder|Pliny]] in the 1st century CE, who apparently had access to writings of late Greek astronomers, and not being an astronomer himself, still correctly identified the [[line of apsides|lines of apsides]] for the five known planets and where they pointed in the zodiac.<ref>{{cite book |author-link=Pliny the Elder |author=Gaius Plinius Secundus |title=Naturalis Historia |trans-title=Natural History |title-link=Natural History (Pliny) |at=Book 2, Chapter 13 |section=An account of the world and the elements: Why the same stars appear at some times more lofty and some times more near |section-url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.02.0137%3Abook%3D2&force=y |access-date=7 August 2014}}</ref> Such data requires the concept of eccentric centers of motion.  

Before around the year 430&nbsp;BCE, [[Meton]] and [[Euktemon]] of Athens observed differences in the [[Season#Astronomical|lengths of the seasons]].<ref name=Evans/> This can be observed in the lengths of seasons, given by equinoxes and solstices that indicate when the Sun traveled 90&nbsp;degrees along its path. Though others tried, Hipparchos calculated and presented the most exact lengths of seasons around 130&nbsp;BCE.

According to these calculations, Spring lasted about {{nobr| {{sfrac|94|1| 2 }} days}}, Summer about {{sfrac|92|1| 2 }}, Fall about {{sfrac|88|1| 8 }}, and Winter about {{sfrac|90|1| 8 }}, showing that seasons did indeed have differences in lengths. This was later used as evidence for the zodiacal inequality, or the appearance of the Sun to move at a rate that is not constant, with some parts of its orbit including it moving faster or slower. The Sun's annual motion as understood by Greek astronomy up to this point did not account for this, as it assumed the Sun had a perfectly circular orbit that was centered around the Earth that it traveled around at a constant speed. According to the astronomer Hipparchos, moving the center of the Sun's path slightly away from Earth would satisfy the observed motion of the Sun rather painlessly, thus making the Sun's orbit eccentric.<ref name=Evans/>

Most of what we know about Hipparchus comes to us through citations of his works by Ptolemy.<ref name=Ptolemy-Almagest/> Hipparchus' models' features explained differences in the length of the seasons on Earth (known as the "first anomaly"), and the appearance of retrograde motion in the planets (known as the "second anomaly"). But Hipparchus was unable to make the predictions about the location and duration of retrograde motions of the planets match observations; he could match location, or he could match duration, but not both simultaneously.<ref>{{cite AV media |series=The New Astronomy |title=Equants, from Part&nbsp;1 of Kepler's ''Astronomia Nova'' |publisher=science.larouchepac.com |url=http://science.larouchepac.com/kepler/astronomianova/part1/5 |access-date=1 August 2014}} — An excellent video on the effects of the equant</ref>

===Ptolemy===
Between Hipparchus's model and Ptolemy's there was an intermediate model that was proposed to account for the motion of planets in general based on the observed motion of Mars. In this model, the deferent had a center that was also the equant, that could be moved along the deferent's line of symmetry in order to match to a planet's retrograde motion. This model, however, still did not align with the actual motion of planets, as noted by Hipparchos. This was true specifically regarding the actual spacing and widths of retrograde arcs, which could be seen later according to Ptolemy's model and compared.<ref name=Evans/>

Ptolemy himself rectified this contradiction by introducing the equant in his writing<ref name=Ptolemy-Almagest>{{cite book |author-link=Ptolemy |first=Claudius |last=Ptolemy |title={{math|Μαθηματικὴ Σύνταξις}} (Mathēmatikē Syntaxis) |title-link=Almagest |trans-title=Mathematical Treatise ("Almagest") |at=IX, 5}}</ref> when he separated it from the center of the deferent, making both it and the deferent's center their own distinct parts of the model and making the deferent's center stationary throughout the motion of a planet.<ref name=Evans/> The location was determined by the deferent and epicycle, while the duration was determined by uniform motion around the equant. He did this without much explanation or justification for how he arrived at the point of its creation, deciding only to present it formally and concisely with proofs as with any scientific publication. Even in his later works where he recognized the lack of explanation, he made no effort to explain further.<ref name=Evans/>

Ptolemy's model of astronomy was used as a technical method that could answer questions regarding astrology and predicting planets positions for almost 1,500&nbsp;years, even though the equant and eccentric were regarded by many later astronomers as violations of pure [[Aristotelian physics]] which presumed all motion to be centered on the Earth. It has been reported that Ptolemy's model of the cosmos was so popular and revolutionary, in fact, that it is usually very difficult to find any details of previously used models, except from writings by Ptolemy himself.<ref name=Evans/>

===From Copernicus to Kepler===
For many centuries rectifying these violations was a preoccupation among scholars, culminating in the solutions of [[Ibn al-Shatir]] and [[Copernicus]]. Ptolemy's predictions, which required constant review and corrections by concerned scholars over those centuries, culminated in the observations of [[Tycho Brahe]] at [[Uraniborg]].

It was not until [[Johannes Kepler]] published his ''[[Astronomia Nova]]'', based on the data he and Tycho collected at Uraniborg, that Ptolemy's model of the heavens was entirely supplanted by a new geometrical model.<ref>{{cite journal |first=Michael |last=Perryman |date=2012-09-17 |title=History of Astrometry |journal=European Physical Journal H |volume=37 |issue=5 |pages=745–792 |arxiv=1209.3563 |bibcode=2012EPJH...37..745P |doi = 10.1140/epjh/e2012-30039-4 | s2cid=119111979}}</ref><ref>{{cite journal |author1=Bracco, C. |author2=Provost, J.-P. |date=24 July 2009 |title=Had the planet Mars not existed: Kepler's equant model and its physical consequences |journal=European Journal of Physics |volume=30 |issue=5 |pages=1085–1092 |doi=10.1088/0143-0807/30/5/015 |arxiv=0906.0484 |bibcode = 2009EJPh...30.1085B |s2cid=46989038}}</ref>