# 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 heavenly bodies. The equant is used to explain the observed speed change in planetary orbit during different stages of the zodiac. 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, 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 speed. However, the planet/center of epicycle will not move uniformly on its deferent.

The reason for the placement of the equant was to maintain uniform circular motion of heavenly bodies, a long standing rule put in place by Aristotle. However, the equant point can be seen as Ptolemy making a compromise with this rule in order to have more accurate predictions for the location of planets.

## 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.[1]

The equant model has a body in motion on a circular path that does not share a center with Earth. The moving objects 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 nonuniform from any 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.[2]

The reason Greek astronomers adopted Ptolemy's model of celestial movement was due to apparent inequalities observed in planetary motion, specifically the movement of Mars, which showed apparent retrograde motion. Mars, Jupiter, and Saturn reach the middle of their retrograde arcs when they are in opposition to the sun, while Mercury and Venus reach the middle of their retrograde arcs when they are in conjunction with the sun. Using the equant, Ptolemy can then account for the lengths of each planets retrograde arcs.

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 while still involving only perfect circular motions.

Ptolemy's model of astronomy was used as a technical method that could answer questions regarding astrology and predicting planets positions.

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.[3]

## Opposition

This concept solved the problem of accounting for the anomalistic motion of the planets but was believed by some to compromise the goals of the ancient astronomer, namely uniform circular motion. The speed of the planet is nonuniform from every point within the orbit, the Earth, the deferent and 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,[4] and Nicolaus Copernicus. Dislike of the equant was a major motivation for Copernicus to construct his heliocentric system.[5][6] The introduction of the equant point meant that Ptolemy was breaking a long standing rule put in place by Aristotle, planetary motions no longer centered around the Earth, the center of the cosmos.[7] This violation of perfect circular motion around the Earth bothered many thinkers, especially Copernicus who mentions the equant as a monstrous construction in De Revolutionibus. However, Copernicus kept the epicycle and deferent in order to make predictions regarding the planets future positions.

## References

1. ^ Eccentrics, deferents, epicycles and equants (Mathpages)
2. ^ Evans, James (1984). "On the function and the probable origin of Ptolemy’s equant". American Journal of Physics (52): 1080.
3. ^ Bracco; Provost (2009). "Had the planet Mars not existed: Kepler's equant model and its physical consequences". European Journal of Physics 30: 1085–92.
4. ^ Craig G. Fraser, 'The cosmos: a historical perspective', Greenwood Publishing Group, 2006 p.39
5. ^ Kuhn, Thomas (1957 (copyright renewed 1985)). The Copernican Revolution. Harvard University Press. pp. 70–71. ISBN 0-674-17103-9.
6. ^ Koestler A. (1959), The Sleepwalkers, Harmondsworth: Penguin Books, p. 322; see also p. 206 and refs therein. [1]
7. ^ Van Helden. "Ptolemaic System". Retrieved 20 March 2014.