# Equant

For the telecommunications company named Equant, see Equant (France Télécom).
The basic elements of Ptolemaic astronomy, showing a planet on an epicycle with a deferent and an equant point.

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 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.

## Placement

The equant point, indicated in the diagram by the large • , is placed so that it is directly opposite the Earth from the center of the deferent (known as the "eccentric"), indicated by the X. A planet or the center of an epicycle (a smaller circle carrying the planet) was conceived to move with a uniform angular speed with respect to the equant. In other words, to a hypothetical observer placed at the equant point, the center of the epicycle would appear to move at a steady angular speed. However, the center of the epicycle will not move at a uniform speed along its deferent.

The reason for the implementation of the equant was to maintain a semblance of uniform circular motion of heavenly bodies, a long standing article of faith originated by Aristotle for philosophical reasons, while also allowing for the best match of the computations of the observed movements of the body.[1]

## Equation

The angle α at the earth between the planet and the equant is a function of time t:

$\alpha(t) = \Omega t - \arcsin\left(\frac{E}{R} \sin(\Omega t) \right)$

where Ω is the constant angular speed seen from the equant which is situated at a distance E when the radius of the deferent is R.[2]

The equant model has a body in motion on a circular path that does not share a center with Earth. The moving object's speed will actually vary during its orbit around the outer circle (dashed line), faster in the bottom half and slower in the top half. The motion is considered uniform only because the planet sweeps around equal angles in equal times from the equant point. The speed of the object is non-uniform when viewed from any other point within the orbit.

## Discovery and use

Ptolemy introduces the equant in Almagest. The evidence for the equant relies mostly based on observations made by himself and an associate named Theon that is mentioned. He asserts that the epicycle's center is carried on a circle, the deferent, whose center is different from the center of uniform motion. Ptolemy never mentions how he arrived at his discovery, however the idea of the equant point was introduced during Ptolemy's explanation of longitudes and equants are later found through direct calculations. These calculations show that the center of the deferent lies halfway between the Earth and the equant point, known to earlier astronomers as the "eccentric".[1]

In models of the universe that precede Ptolemy, generally attributed to Hipparchus, the eccentric and epicycles were already a feature. The Roman 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.[3] Such data requires the concept of eccentric centers. Most of what we know about Hipparchus comes to us through mentions of his works by Ptolemy in the Almagest. These model 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"). Hipparchus was unable to make the predictions about the location and duration of retrogrades match observations; he could match location, or he could match duration, but not both simultaneously.[4] It somehow occurred to Ptolemy (he doesn't explain how it came about), that separation of the center of uniform motion and the center of the deferent, both already separated from the Earth, could solve the problem quite accurately.[1]

The equant was the third of the basic constructions of the Ptolemaic model of planetary motion. The first part of the model is the eccentric. The eccentric places Earth slightly off-center of the planet's path. This can be seen in the image above, shown by the center, X, and the Earth underneath it. The second part of the model is the epicycle. This is illustrated in the image by the orange planet orbiting the small circle while moving along its larger orbit. When the speed and direction of the planets rotation were applied annual retrograde motion could be accounted for. These two constructs alone did not form a satisfactory model of planetary motion, so a third part was added to the model, the equant. This resulted in the geometric center, X, being replaced by the equant,  • , as the center of motion. This model closely approximated the planetary motions, including those of the retrograde, while still involving only perfect circular motions, uniform, when properly interpreted.

Ptolemy's model of astronomy was used as a technical method that could answer questions regarding astrology and predicting planets positions for almost 1500 years, not completely replaced until Johannes Kepler published his Astronomia Nova. Until then, this technical astronomy was held as a field of mathematics, quite separate from study of the true nature and meaning of the universe, which was a branch of philosophy.[5]

Kepler also had great use for the equant point. According to Kepler himself, the equant was a cornerstone of his work. He used the equant parameters to derive accurate distances for the sun and Mars. These calculations also led to the abandonment of the equant model because the calculated distances did not fit with previously believed distances. Had Mars not been present however, the equant would have continued to be used in the planetary motion model for far longer.[6]

## Criticism

The equant solved the last major problem of accounting for the anomalistic motion of the planets but was believed by some to compromise the principles of the ancient Greek philosopher/astronomers, namely uniform circular motion. The uniformity was generally assumed to be observed from the center of the deferent, and since that happens at only one point, only non-uniform motion is observed from any other point. Ptolemy moved the observation point explicitly off the center of the deferent to the equant. This can be seen as breaking part of the uniform circular motion rules. Noted critics of the equant include the Persian astronomer Nasir al-Din Tusi who developed the Tusi-couple as an alternative explanation,[7] and Nicolaus Copernicus, whose alternative was a new pair of epicycles for each deferent. Dislike of the equant was a major motivation for Copernicus to construct his heliocentric system.[8][9] The introduction of the equant point meant that Ptolemy was breaking a long standing rule put in place by Aristotle, planetary motions no longer were uniform around the Earth, the center of the cosmos.[10] This violation of perfect circular motion around the center of the deferent bothered many thinkers, especially Copernicus who mentions the equant as a monstrous construction in De Revolutionibus. Copernicus' movement of the Earth away from the center of the universe obviated the primary need for Ptolemy's epicycles by explaining retrograde movement as an optical illusion, but he re-instituted two smaller epicycles into each planet's motion in order to replace the equant.

## References

1. ^ a b c Evans, James (1984). "On the function and the probable origin of Ptolemy’s equant". American Journal of Physics (52): 1080.
2. ^ Eccentrics, deferents, epicycles and equants (Mathpages)
3. ^ Pliny the Elder. The Natural History, Book 2: An account of the world and the elements, Chapter 13: Why the same stars appear at some times more lofty and some times more near. Retrieved 201 August 7.
4. ^ "The New Astronomy - Equants, from Part 1 of Kepler's Astronomia Nova". science.larouchepac.com. Retrieved 2014 August 1. An excellent video on the effects of the equant
5. ^ Perryman, Michael (2012 September 17). History of Astrometry. Retrieved 2014 August 7.
6. ^ Bracco; Provost (2009). "Had the planet Mars not existed: Kepler's equant model and its physical consequences". European Journal of Physics 30: 1085–92. doi:10.1088/0143-0807/30/5/015.
7. ^ Craig G. Fraser, 'The cosmos: a historical perspective', Greenwood Publishing Group, 2006 p.39
8. ^ Kuhn, Thomas (1957 (copyright renewed 1985)). The Copernican Revolution. Harvard University Press. pp. 70–71. ISBN 0-674-17103-9. Retrieved 2014 August 5.
9. ^ Koestler A. (1959), The Sleepwalkers, Harmondsworth: Penguin Books, p. 322; see also p. 206 and refs therein. [1]
10. ^ Van Helden. "Ptolemaic System". Retrieved 20 March 2014.