In classical mechanics though the Earth going round the Sun is not a truth. It is just more convenient for the math.Bingo! The only ancient who posited anything other than the earth as the center of the universe was Pythagoras, to whom Copernicus dedicated his De Revolutionibus Orbium Coelestium.
Actually, the centrifugal force associated with the Sun rotating around the Earth would be HUGH! Even Einstein can't help you here. Newton had the Earth and Sun rotating around their common center of Gravity.
Kostler gives more or less the same view as this author.
Koestler also claims that the Galileo's "Dialogues Concerning Two Sciences" was bad astronomy and bad physics. For instance, he explained the tides are resulting from the Earth's rotation, rather than - as Kepler urgently suggested - the effect of the Moon's gravity. Galileo's contribution was in the inventing the science of dynamics.
Ptolemic astronomy worked as well or better than Coperican for predicting the positions of the planets and was simplier to apply. It also fulfilled the intuitive need for a mechanism - the nested crystalline spheres - to carry the planets along. Until Tycho's painstaking observations, no one had any better model for planetary motion consistent with observations.
Kepler was captivated by the smoother trajectories that the Copernican model afforded. Ptolemic trajectories superimposed the Earth's orbit on top of the orbits of the planets, causing a funny spiral trajectory, compared to the oval shapes resulting from Copernius' epicycles. When Kepler attempted to fit Tycho's observations of Mars to the Copernican model, he found they would not comply and hit upon the ellipse.
Newton's Laws perfectly explain Kepler's orbits and a great number of other things as well. (Such as tides and the precession of the equinox.)
It wasn't until Maxwell invented electrodynamics, that inconsistencies between E/M and mechanics lead to relativity. (The precession of the perihelion of Mercury had been noticed prior to the theory of relativity, but did not lead to it. Rather, it is one of the three classical tests of General Relativity. The other were only observed afterwards: Gravitational red shift and Deflection of starlight during an eclipse.)
I seem to remember that in the forward to Copernicus's book, he says just that (I believe the experts are still debating whether he meant it to deflect criticism from Rome).