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# Equant

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 Title: Equant Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Equant

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.

## Contents

• Placement 1
• Equation 2
• Discovery and use 3
• Criticism 4
• References 5

## 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.[1]

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.

## 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 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).[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 of motion. Most of what we know about Hipparchus comes to us through mentions of his works by Ptolemy in the 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.[4] Ptolemy's introduction of the equant resolved that contradiction: the location was determined by the deferent and epicycle, while the duration was determined by uniform motion around the equant.

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, even though the equant and eccentric were violations of pure Uraniborg.

It wasn't until

• Equidimensional: This is a synonym for equant when it is used as an adjective.