Equant

, showing a planet on an epicycle (smaller dashed circle), a deferent (larger dashed circle), the eccentric (×) and an equant (•). Equant (or punctum aequans) 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 different stages of the planetary 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. Ptolemy does not have a word for the equant – he used expressions such as "the eccentre producing the mean motion".

, showing a planet on an epicycle (smaller dashed circle), a deferent (larger dashed circle), the eccentric (×) and an equant (•).]] Equant (or punctum aequans) 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 different stages of the planetary 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. Ptolemy does not have a word for the equant – he used expressions such as "the eccentre producing the mean motion". \alpha = \Omega t - \arcsin\left(\frac{E}{\ R\ }\ \sin(\Omega t) \right) where is the constant angular speed seen from the equant which is situated at a distance when the radius of the deferent is . ==Discovery and use== Ptolemy introduced the equant in "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). ===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 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 lines of apsides for the five known planets and where they pointed in the zodiac. Such data requires the concept of eccentric centers of motion. Before around the year 430 BCE, Meton and Euctemon of Athens observed differences in the lengths of the seasons. This can be observed in the lengths of seasons, given by equinoxes and solstices that indicate when the Sun traveled 90 degrees along its path. Though others tried, Hipparchos calculated and presented the most exact lengths of seasons around 130 BCE. According to these calculations, Spring lasted about , Summer about , Fall about , and Winter about , 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. Most of what we know about Hipparchus comes to us through citations of his works by Ptolemy. 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. ===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. Ptolemy himself rectified this contradiction by introducing the equant in his writing 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. 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 recognised the lack of explanation, he made no effort to explain further. 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 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. ===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. ==See also== Equidimensional: This is a synonym for equant when it is used as an adjective. ==External links== Ptolemaic System – at Rice University's Galileo Project. Java simulation of the Ptolemaic System – at Paul Stoddard's Animated Virtual Planetarium, Northern Illinois University.

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