Ptolomy - Equants and the Amagest

Equant

Not much new theory came after Hipparchus until Ptolemy (2nd century AD).

Ptolemy introduced another new geometric concept - the 'equant'. In this model the planet moves on a circle. The Earth is placed some distance from the centre of the circle at the eccentre. The equant is placed symmetrically opposite the earth on the other side of the centre.

The planet moves round the circle such that it's apparent motion seen from the equant is uniform. To do this, it must in fact speed up and slow down to account for its varying distance from the equant. In practice this is getting to look a bit like an ellipse with the eccentre and equant being the two focus points. However, the shape of the "orbit" is still a circle.

Almagest

Ptolemy went on to write a  major document that became known as the Almagest. In it he used a combination of  eccentres, equants and epicycles to represent the movement of all the planets, including their variation in speed and retrograde motion.

This diagram shows the movement of an outer planet as described in the Almagest. The planet moves round an epicycle, the centre of which moves round a referent circle. Earth and an equant point are placed an appropriate distance from the centre of the referent circle. The centre of the epicycle moves such that it would be in uniform motion seen from the equant point. 

The "Mean Sun" moves round the centre of the referent circle with uniform motion and the line from the centre of the epicycle to the planet is maintained parallel to the line from the earth to the Sun.  Ptolemy's Almagest was a major step forward in providing the means to predict the positions of planets and it remained the most complete theory for the next 1000 years.

Cappella

One of the few new ideas put forward in Medieval times is the theory that Mercury and Venus must revolve round the Sun. This was documented by Martianus Cappella (4th century AD) although the idea may go back much further. 

In this diagram the Earth is at the centre. The Sun and Moon orbit the Earth. Mercury and Venus are interlinked in orbits round the Sun and seem to be able to overlap.

All the outer planets move round the Earth in epicycles.

Note also that the Sun's orbit is not central, but is offset to an eccentre.