  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.
