Optimizing the Ptolemaic Model of Planetary and Solar Motion

Ptolemy-s planetary model is an ancient geocentric astronomical model, describing the observed motion of the Sun and the planets. Ptolemy accounted for the deviations of planetary orbits from perfect circles by introducing two small and equal shifts into his model. We show that Ptolemy-s choice of shifts allowed him to approximate the true eccentricity of planetary orbits only in the first order, linear in eccentricity. We show that, if the shifts were tuned in the ratio 5/3, the model precision could be improved substantially, perhaps delaying its rejection. The best achievable precision is quadratic in eccentricities. Although, to achieve it fully, one would have to introduce Ptolemaic shifts in the epicycles as well as in deferents the planets.