Ptolemy and Mathematical TechniquesPtolemy did not so much introduce new mathematical techniques as modify existing ones to suit his purpose. Much advanced work had already been done in mathematical astronomy prior to Ptolemy: the epicyclic and eccentric models of planetary motion were long established, and their equivalence proved by Apollonius of Perga (c.200 BC). Hipparchus had employed his own observations and those of the Babylonians in constructing models of lunar and solar motion. Plane and spherical trigonometry had already been developed.
However, in one specific area, the geometrical modelling of the motions of the planets other than the sun and moon, he was a pioneer, willing to break with tradition, or at least deform it. His originality here lay in the introduction of a geometrical device later named the equant point. In this construction, the planet moved around the earth in an eccentric circle (that is, one not centred on the earth). The equant point lay symmetrically opposite the centre of the eccentric circle from the earth, and it was from this point that the motion of the planet would appear uniform.
The equant point stretched the traditional requirement of only uniform circular motions in the heavens to its limit, and proved philosophically unpalatable for many Islamic and western medieval astronomers, and thus stimulated subsequent work, especially among the Islamic astronomers, to find alternative models.