Although Kepler adopted the heliocentric approach of Copernicus,
what he effectively first did was to perfect Ptolemy's model of the solar system (or, rather,
its heliocentric equivalent). Thus, Kepler replaced Ptolemy's
erroneous equantless model of the sun's apparent orbit around the
earth with a corrected version containing an equant--in the process,
halving the eccentricity of the orbit (see Fig. 1). Kepler also introduced equants
into the epicycles of the superior and inferior planets. Once he had perfected Ptolemy's model, the heliocentric nature of
the solar system became manifestly apparent to Kepler. For instance,
he found that the epicycles of the superior planets, the sun's
apparent orbit around the earth, and the deferents of the inferior
planets all have exactly the *same* eccentricity. The obvious implication
is that these circles all correspond to some common motion within the
solar system--in fact, the motion of the earth around the sun.

Once Kepler had corrected the Almagest model, he compared its predictions
with his observational data. In particular, Kepler investigated
the apparent motion of Mars in the night sky. Kepler found that his model performed extremely well,
but that there remained small differences between its predictions and the observational data. The maximum discrepancy was about : *i.e.*, about the apparent size of the
sun. By the standards of naked-eye astronomy, this was a very small discrepancy indeed. Nevertheless, given the incredible accuracy of Tycho Brahe's
observations, it was still significant. Thus, Kepler embarked on an epic new series
of calculations which eventually lead him to the conclusion that the
planetary orbits are actually eccentric ellipses, rather than eccentric circles.
Kepler published the results of his research in *Astronomia
Nova* (New Astronomy) in 1609 CE.
It is
interesting to note that had Tycho's data been a little less accurate,
or had the orbit of Mars been a little less eccentric, Kepler might well
have settled for a model which was kinematically equivalent to
a perfected version of the model described in the Almagest. We can also
appreciate that, given the far less accurate observational data available to
Ptolemy, there was no way in which he could have discerned the
very small difference between elliptical planetary orbits and the eccentric
circular orbits employed in the Almagest.