Compare the way ptolemy provided proof for his theory with the way Kepler provided proof for this theory?

Ptolemy provided proof for his geocentric theory through a complex system of epicycles, deferents, and observations of planetary motion, relying heavily on empirical data and geometric models to match observed celestial movements. In contrast, Kepler, working with the data collected by Tycho Brahe, derived his heliocentric theory using mathematical laws, specifically his three laws of planetary motion, which demonstrated that planets move in elliptical orbits with varying speeds. While Ptolemy's approach was largely qualitative and geometric, Kepler's was quantitative and focused on deriving accurate mathematical relationships from observational data. This shift marked a significant evolution in the methodology of astronomical proof.