Editing Deferent and epicycle (section)

==Introduction==
[[File:Ptolemaic elements.svg|thumb|The basic elements of Ptolemaic astronomy, showing a planet on an epicycle (smaller dashed circle), a deferent (larger dashed circle), the eccentric (×) and an [[equant]] (•).]]
In both Hipparchian and Ptolemaic systems, the [[planet]]s are assumed to move in a small circle called an ''epicycle'', which in turn moves along a larger circle called a ''deferent'' (Ptolemy himself described the point but did not give it a name<ref>See page 21 of the Introduction in {{cite book |translator-link=Gerald J. Toomer |translator-first=Gerald J. |translator-last=Toomer |title=Ptolemy's Almagest |date=1984 |url=https://classicalliberalarts.com/resources/PTOLEMY_ALMAGEST_ENGLISH.pdf}}</ref>). Both circles rotate eastward and are roughly parallel to the plane of the Sun's apparent orbit under those systems ([[ecliptic]]). Despite the fact that the system is considered [[geocentric]], neither of the circles were centered on the earth, rather each planet's motion was centered at a planet-specific point slightly away from the Earth called the ''eccentric''. The [[orbit]]s of planets in this system are similar to [[epitrochoid]]s, but are not exactly epitrochoids because the angle of the epicycle is not a linear function of the angle of the deferent.

In the Hipparchian system the epicycle rotated and revolved along the deferent with uniform motion. However, Ptolemy found that he could not reconcile that with the Babylonian observational data available to him; in particular, the shape and size of the apparent retrogrades differed. The angular rate at which the epicycle traveled was not constant unless he measured it from another point which is now called the ''[[equant]]'' (Ptolemy did not give it a name). It was the angular rate at which the deferent moved around the point midway between the equant and the Earth (the eccentric) that was constant; the epicycle center swept out equal angles over equal times only when viewed from the equant. It was the use of equants to decouple uniform motion from the center of the circular deferents that distinguished the Ptolemaic system. For the outer planets, the angle between the center of the epicycle and the planet was the same as the angle between the Earth and the Sun.

Ptolemy did not predict the relative sizes of the planetary deferents in the ''Almagest''. All of his calculations were done with respect to a normalized deferent, considering a single case at a time. This is not to say that he believed the planets were all equidistant, but he had no basis on which to measure distances, except for the Moon. He generally ordered the planets outward from the Earth based on their orbit periods. Later he calculated their distances in the ''[[Planetary Hypotheses]]'' and summarized them in the first column of this table:<ref>{{cite journal |title=The Structure and Function of Ptolemy's Physical Hypotheses of Planetary Motion |last=Andrea |first=Murschel |journal=Journal for the History of Astronomy |issue=xxvii |pages=33&ndash;61 |year=1995 |volume=26 |doi=10.1177/002182869502600102 |url=https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1995JHA....26...33M |access-date=2 August 2014 |bibcode=1995JHA....26...33M|s2cid=116006562 }}</ref>

{| class="wikitable" style="text-align:center"
 |+ Ptolemy's estimates of orbit sizes
 |-
 ! Body
 ! Mean size <br /><small>(in Earth radii)</small>
 ! Modern value <br /><small>([[semimajor axis]], <br />in Earth radii)</small>
 ! Ratio <br /><small>(modern/Ptolemy)</small>
 ! Ratio <br /><small>(modern/Ptolemy, <br />normalized to Sun = 1)</small>
 |-
 ! Moon
 | {{0|00,0}}48{{0|.0}}
 | {{0|000,0}}60.3
 | {{0}}1.26
 | 0.065
 |-
 ! Mercury
 | {{0|00,}}115{{0|.0}}
 | {{0|00}}9,090{{0|.0}}
 | 79.0{{0}}
 | 4.1{{0|00}}
 |-
 ! Venus
 | {{0|00,}}622.5
 | {{0}}16,980{{0|.0}}
 | 27.3{{0}}
 | 1.4{{0|00}}
 |-
 ! Sun
 | {{0}}1,210{{0|.0}}
 | {{0}}23,480{{0|.0}}
 | 19.4{{0}}
 | 1{{0|.000}}
 |-
 ! Mars
 | {{0}}5,040{{0|.0}}
 | {{0}}35,780{{0|.0}}
 | {{0}}7.10
 | 0.37{{0}}
 |-
 ! Jupiter
 | 11,504{{0|.0}}
 | 122,200{{0|.0}}
 | 10.6{{0}}
 | 0.55{{0}}
 |-
 ! Saturn
 | 17,026{{0|.0}}
 | 225,000{{0|.0}}
 | 13.2{{0}}
 | 0.68{{0}}
 |-
 ! Star shell
 | 20,000{{0|.0}}
 | {{n/a}}
 | {{n/a}}
 | {{n/a}}
 |}
Had his values for deferent radii relative to the Earth–Sun distance been more accurate, the epicycle sizes would have all approached the Earth–Sun distance. Although all the planets are considered separately, in one peculiar way they were all linked: the lines drawn from the body through the epicentric center of all the planets were all parallel, along with the line drawn from the Sun to the Earth along which Mercury and Venus were situated. That means that all the bodies revolve in their epicycles in lockstep with Ptolemy's Sun (that is, they all have exactly a one-year period).{{citation needed|date=August 2014}}

Babylonian observations showed that for [[superior planet]]s the planet would typically move through in the night sky slower than the stars. Each night the planet appeared to lag a little behind the stars, in what is called [[prograde motion]]. Near [[opposition (planets)|opposition]], the planet would appear to reverse and move through the night sky faster than the stars for a time in [[apparent retrograde motion|retrograde motion]] before reversing again and resuming prograde. Epicyclic theory, in part, sought to explain this behavior.

The [[inferior planet]]s were always observed to be near the Sun, appearing only shortly before sunrise or shortly after sunset. Their apparent retrograde motion occurs during the transition between evening star into morning star, as they pass between the Earth and the Sun.