Ptolemy proposed a geocentric model, with Earth at the center and planets orbiting it. Copernicus suggested a heliocentric model, with the Sun at the center of the solar system. Galileo's telescopic observations supported the heliocentric model and provided evidence for Copernicus' theory.
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Nicolaus Copernicus found out what happens with the Earth's movements.
The main difference was that Ptolemy's model was geocentric (Earth-centred) and Copernicus's was heliocentric (Sun-centred). Ptolemy's model came from ancient times while Copernicus's was much later (1543). Both models represented the planets' orbits by using combinations of circles and epicycles to explain the way the planets move among the stars. Copernicus found that the orbits of the inner planets could be explained more simply. That is to say that the epicycles used for all the orbits were smaller, and for the inner planets a lot smaller. Both models represented the planets' positions with reasonable accuracy given the crude observational methods used in those days. Until gravity and the laws of dynamics were discovered about 150 years after the publication of Copernicus's system, there was no way of deciding which model was the 'right' one.
Both Copernicus heliocentric and the Ptolemaic models agreed on the need for epicycles. These were miniature orbits that the celestial bodies travelled on as well as their normal orbits.Copernicus still invigaed the bodies orbiting in perfect circles and had to put these in to explain some of the movements of the planets. This may have contributed to his ideas being largely dismissed as it was just as complicated as the already established model. It was not until Kepler proposed elliptical orbits was this problem resolved.
Yes, Galileo rejected the notion of geocentrism and instead supported the heliocentric model proposed by Copernicus. Galileo's observations through his telescope provided evidence that supported a sun-centered solar system.
Galileo did not prove that Earth was not flat (It had already been agreed that it was round). Galileo invented the telescope and used it to prove that there were objects which did not orbit the Earth, supporting Copernicus' Heliocentric model for the universe. The Earth was known to be round by the ancient Greeks, but I do not who it found out initially (sorry).