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 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.

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, particularly in the size of the apparent retrograde motion of all solar system bodies except the sun and the moon.

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

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.

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.

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).

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