Editing Equant (section)

== Placement ==
The '''equant point''' (shown in the diagram by the large&nbsp;•&nbsp;), is placed so that it is directly opposite to Earth from the [[deferent and epicycle|deferent]]'s center, known as the ''eccentric'' (represented by the&nbsp;×&nbsp;). A [[planet]] or the center of an [[epicycle]] (a smaller circle carrying the planet) was conceived to move at a constant [[angular speed]] with respect to the equant. To a hypothetical observer placed at the equant point, the epicycle's center (indicated by the small&nbsp;·&nbsp;) would appear to move at a steady angular speed. However, the epicycle's center will not move at a constant speed along its deferent.<ref name="Evans">{{cite journal| first=James| last=Evans| title=On the function and probable origin of Ptolemy's equant| date=April 18, 1984| url=http://www2.ups.edu/faculty/jcevans/Equant.pdf| access-date=August 29, 2014| journal=[[American Journal of Physics]]| volume=52| number=12| pages=1080–89| doi=10.1119/1.13764| bibcode=1984AmJPh..52.1080E}}</ref>

=== Motivation ===
The reason for the implementation of the equant was to maintain a semblance of constant [[diurnal motion|circular motion]] of [[astronomical object|celestial 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 bodies, particularly in the size of the [[apparent retrograde motion]] of all [[Solar System]] bodies except the [[Sun]] and the [[Moon]].

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

Applied without an epicycle (as for the Sun), using an equant allows for the angular speed to be correct at perigee and apogee, with a ratio of <math>(1+e)^2/(1-e)^2</math> (where <math>e</math> is the [[orbital eccentricity]]). But compared with the [[Keplerian orbit]], the equant method causes the body to spend too little time far from the Earth and too much close to the Earth. For example, when the [[eccentric anomaly]] is π/2, the Keplerian model says that an amount of time of <math>\pi/2-e</math> will have elapsed since perigee (where the period is <math>2\pi</math>, see [[Kepler equation]]), whereas the equant model gives <math>\pi/2-\arctan(e),</math> which is a little more. Furthermore, the [[true anomaly]] at this point, according to the equant model, will be only <math>\pi/2+\arctan(e),</math> whereas in the Keplerian model it is <math>\pi/2+\arcsin(e),</math> which is more. However, for small eccentricity the error is very small, being [[asymptotic]] to the eccentricity to the third power.

=== Equation ===
The angle {{mvar|α}} whose vertex is at the center of the deferent, and whose sides intersect the planet and the equant, respectively, is a function of time {{mvar|t}} as follows:

: <math> \alpha = \Omega t - \arcsin\left(\frac{E}{\ R\ }\ \sin(\Omega t) \right) </math>

where {{mvar|Ω}} is the constant angular speed seen from the equant which is situated at a distance {{mvar|E}} when the radius of the deferent is {{mvar|R}}.<ref>See equation 8 in {{cite web |title=Eccentrics, deferents, epicycles and equants |website=Mathpages |url=http://www.mathpages.com/home/kmath639/kmath639.htm}} The quthor derives this in a complicated way using derivatives and integrals, but in fact it follows directly from equation 3.</ref>