Solar System: Ptolemy vs. Copernicus
Ptolemy's model, accepted as true for over a thousand years, is based on common sense: the Earth doesn't move in some orbit. It then follows from simple observation that the Sun does move, in a circle around the Earth. The(more complicated) motion of Mars against the distant starry background can be well explained by taking it to circle around a point which is itself circling around the fixed Earth (this is called "epicyclic motion").Notice the Earth is the center of everything.
We now know better: but the Earth-Mars direction is the same in the Ptolemy's model and ours at all times! Look at the animation to see this. This means that tracking the motion of Mars against the background of stars is no help in deciding which model is correct. This is equally true of all the planets.