Appolonius and Hipparchus - Eccentres and Epicycles
The challenge was to identify a geometry that would reproduce the movement of the Sun, Moon and planets accurately. Two new concepts were introduced - Eccentricity and Epicycles.
An approach studied by Appollonius of Perga (~200BC) was to employ the concept of eccentric circles.
The 'planet' (e.g. the Sun) moves in a circle with uniform speed. The earth is placed some distance from the centre of the circle. From the Earth, the object appears to speed up and slow down. By selecting the best rotation period and eccentre distance, a better approximation of the motion of the Sun or Moon can be achieved than was possible with an Earth centred view.
Today we know that orbits are ellipses rather than circles. Appolonius' ecentre does in fact give some partial compensation for the eccentricity of the Earth's orbit. However, many people remained against the use of the eccentre because "Earth must be and obviously is the centre of the cosmos."
Another alternative, studied by Appollonius, was the concept of 'epicycle'. In this model, the Earth is at the centre of a circle called the 'referent' circle. A secondary circle, called the 'epicycle' moves such that its centre rotates round the referent circle with uniform motion.
The planet (say Mars) rotates round the epicycle with uniform speed.
The rotation periods and relative sizes of the two circles can be selected to give a reasonable approximation of the apparent motion of the outer planets, including the 'retrograde' motion.
In addition, the variation in distance between the planet and Earth could (to some degree) explain the variation in brightness of the planet.
Hipparchus' Cosmic Theory
Hipparchus (~130BC) combined Babylonian observational data with the concepts of eccentre to provide a more accurate description of the movement of the Sun.
He observed that it takes a different number of days for the sun to travel through each of the 'quarters' of the year from solstice to equinox and equinox to solstice.
He worked out that, to reproduce this effect the eccentre should be placed at 1/24th of the radius of the circle and set such that a line to the centre makes 65.5 degrees with a line to the Spring equinox. This gave a reasonable approximation of the observed motion of the Sun.
Hipparchus also devised a model for the Moon using an epicycle, although this was less accurate than his model for the Sun.