I discuss the problem of secular inequalities in Kepler by giving account of a manuscript note that has not been published until 1860. In his note Kepler points out the need for a model, clearly inspired by the method of epicycles, that describes the secular inequalities as periodic ones. I bring attention to this point, that seems to have been underestimated, since the references to Kepler's work usually report only that he observed a decreasing mean motion for Saturn and an increasing one for Jupiter.Kepler is widely credited with abolishing the evils of epicycles in favor of elliptical orbits, but I did not know that he used epicycles himself. The paper starts:
The discovery by Kepler of the elliptic shape of the planetary orbits is often consideredEpicycles are widely mocked as being contrary to science and rational thought, but they are legitimate tools for representing orbits.
as a discontinuity with the traditional models of Classical Astronomy, based on geometrical tools such as circles, epicycles and equants. The enthusiastic announcement of the discovery in chapter LVI of Astronomia Nova , where Kepler says that he was “quasi e somno expergefactus, et novam lucem intuitus” (like suddenly awakened from sleep, and seeing a new light) seems to confirm that this was his feeling, too. But, as often happens, reality turned out to be complex enough to escape our theories: the orbits of the planets are not exactly elliptic. This is well known today, but it seems that the circumstancy that long term deviations from the elliptic motion have been investigated in great detail by Kepler himself, with an explicit conjecture that these deviations should be periodic, is not so known, even among astronomers.
You might say that the planetary orbits, as seen from the Earth, are circles. The first order correction is the Ptolemy epicycles, related to the Earth's orbit around the Sun. The second order correction is the Copernican epicycles, related to the elliptical orbits, The third order correction is the Kepler epicycles, related to the gravity of other planets. Modern representations might use a Fourier series with hundreds of terms.