Ptolemy's Venus

The main feature of the Inferior planets (Venus and Mercury) is that they never appear at opposition to the Sun. They oscillate to the east and west of the Sun, appearing as morning or evening "stars".

Their periods are:

• The mean tropical period – return to the same longitude around the ecliptic.
• The mean synodic (anomalistic) period – return to same (east or west) elongation.

The periods are not fixed but we can establish mean values from long-term observations. These planets may be east or west of the Sun but their average position is the same as the Sun so their mean motion in longitude (tropical period) is the same as the mean Sun. This lead some early astronomers to suspect that Mercury and Venus orbit the Sun (which in turn orbits Earth). However, Ptolemy does not mention this possibility and sticks to his constructs of equants, deferents and epicycles.

The coordinate system is longitude and latitude relative to the ecliptic with the vernal equinox ϒ as the reference point.

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 mean Sun λm☉ revolves around Earth in one Tropical Year.

An Equant circle of radius R is drawn with centre E, distant 2e from the Earth in the direction of λa.

• λa is believed to be fixed relative the the stars so increases by 1 degree per century relative to the vernal equinox.
• The point S rotates round the Equant circle with uniform motion, west-to-east, with a period equal to the mean tropical period of the planet.(In this case, the same as the Sun)

A deferent circle of radius R is drawn with centre D, distant e from Earth in the direction λa.

• The centre of an epicycle C travels around the deferent circle such that its motion, as seen from the centre of the Equant circle is uniform. C and S are very close together and may be difficult to separate in the diagram.
• Thus C does not rotate round the deferent circle with uniform motion relative to the deferent centre. This is “breaking the rules” of Greek philosophy but Ptolemy does not seem to worry.
• “Bisection of Eccentricity” (using D and E spaced 1e and 2e from Earth ) is introduced in the Almagest, but Ptolemy does not explain how he decided on this construct.
• The use of equant and deferent circles seeks to replicate the variation of apparent angular velocity. As the orbit of Venus has low eccentricity, the equant and deferent circles are close together.

An epicycle is drawn with centre C and radius r.

• The planet moves round the epicycle, west-to-east, at a uniform rate of the mean anomalistic period of the planet relative to the vector E->C.
• This results in the planet appearing to the east and west of the Sun as viewed from Earth.

To account for variations in latitude:

• The deferent circle is tilted by 0.166 degrees and the epicycle is tilted by a further 3.5 degrees.
• The tilt is along the line of the ascending node, Ω, which is deemed to be fixed relative to the stars.

For Venus, R=60, e=1.25, r=43.1666 partes.

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).