The model goes nuts when we have to deal with the retrograde motion of Mars, however, which, before Galileo, had been explained with a looping orbit. Elliptical orbits, too, had to be explained with much more complex motion on the part of the rest of the observable universe.
When G was first hauled before the church, in fact, he skated by saying that the heliocentric theory was simply a model he had constructed to make his calculations easier, as it could use simpler means to express the complex movements of the heavens.
Mars=Mercury... I think.