Who used mathematics to prove the theories of Copernicus?

Answer 1

In physics, theories are not mathematically proved but instead they are compared with physical observations to see if they are correct. Copernicus's theory was not just an idea that the Sun is at the centre, but also a detailed mathematical system of circle and epicycles that provided a model of the planets' orbits. At the time of Copernicus and for a time afterwards, his theory was used to predict the planets' positions, and when the predictions were found to be 'correct', the theory was considered 'proved' .

But later Tycho Brahe developed more accurate measurement techniques, and using those he discovered small errors in the predicted positions of the planets. That led to Kepler's work and the discovery of elliptical orbits.