Ptolemy's model of the Moon's motion is based on the work of Hipparchus but with
extra features related to "evection" and "prosneusis of the epicycle".
The Moon revolves around the Earth, but its motion is quite irregular:
The Moon's orbit is eccentric and this causes its apparent angular velocity to vary.
The amount of eccentricity varies during the month due to the gravitational influence of the
Sun. This secondary effect is called “evection” and was discovered by Ptolemy.
The variations in angular velocity are not the same each month.
The Moon's orbit is inclined to the ecliptic by about 5 degrees, so we need to
calculate latitude as well as longitude. The direction of
the "tilt" (ascending node) changes from month to month.
We can identify several different "periods" for the Moon's orbit:
Synodic Month: time from New Moon to New Moon.
Tropical Month: time to return to the same longitude along the ecliptic.
Anomalistic Month: time to return to the same angular velocity.
Draconic Month: time return to the same latitude relative to the ecliptic.
None of these periods are consistent month-to-month. We may only establish their mean values.
Ptolemy obtained these mean values with great accuracy using observations of Lunar Eclipses
spanning nearly 800 years.
In the diagram we see:
The Earth is in the centre of the Universe and consists of the “spheres” of earth, water, air and fire,
as described by Aristotle.
The ecliptic is the reference plane and the vernal equinox ϒ is the reference point.
Coordinates are described in terms of longitude (relative to ϒ )
and latitude relative to the ecliptic.
The Ecliptic Mean Sun λm☉ revolves around Earth, with uniform angular
velocity, west-to-east, once per tropical year, .
The mean Moon λm☾ revolves around Earth, with uniform angular velocity relative
to the mean Sun, west-to-east, once per synodic month.
The line of apsides λa rotates round Earth,
with uniform retrograde angular velocity relative to the mean Sun, once per synodic month.
An eccentric D is placed at distance e from the Earth
along the line of apsides λa.
A deferent circle is placed with radius R, centred on D.
The centre of an epicycle C travels around the deferent circle, west-to-east,
once per synodic month with motion that is uniform as seen from Earth.(I.e., in the direction
of the mean Moon λm☾.)
An epicycle is drawn with radius r, centred on C.
With this arrangement:
The epicycle centre C moves closer to, and further from, Earth thus modifying
its effect on the apparent angular motion at different times of the month.
The vector from Earth to C is in uniform motion, so rotation around the deferent
circle cannot be uniform relative to its centre D. That goes against the principles
of Greek philosophy, but Ptolemy does not seem to worry and makes no comment about it.
Uniform rotation around the epicycle does not satisfy the the observed positions of
the Moon at the octants (half way between the quarters), so an additional construct is used, knowns as
"prosneusis of the epicycle":
The point D' is established directly opposite D.
The Moon travels around the epicycle, east-to-west, at an angular rate equal to its mean anomalistic
period, relative to the direction of the vector D'C.
And finally, to account for variations of latitude:
The entire structure is projected onto a plane set at an angle of 5 degrees to the ecliptic.
The direction of the tilt (ascending node Ω ) migrates by 0.053
degrees per day east-to-west.
This is not a true representation of inclination, but that is the way Ptolemy handled variations in latitude
and it does not introduce much error.
Ptolemy established the values R=49.6833 and e=10.3166
and r=5.25. These values are in arbitrary units or "partes" that have no
physical meaning.
The animation starts at noon on the 1st day of the month of Thoth in the first year of the reign of Babylonian King
Nabonassar.(Feb 26th 747BC, JD 1448273.0).