As you may or may not know, Copernicus was really the father of the heliocentric theory. Galileo expanded on his model later on but Copernicus came up with his model on his own. Copernicus decided that the Earth didn't have to be at the center of the universe and that it made much more sense to put the sun there. By looking at all the figures he came up with the order of the planets and the fact that the moon revolved around the Earth as shown in the model below.
Remember that problem though before with the planets moving backwards and getting brighter? His model solved this problem easily and without any epicycles needed to show it. The animation below shows this.
It is widely believed though that Copernicus did away with Ptolemy's epicycles all together and this is not true. If you notice, in Copernicus' model the planets still orbited in a complete circle while we know they really orbit in ellipses. Copernicus needed to use some epicycles to explain this problem but he used much fewer than Ptolemy used.
This heliocentric theory has been worked on and perfected for centuries until it is known as what we learn in school today. Through many adaptations, many problems, and many solutions Eudoxus' geocentric theory slowly made it's way to be changed into the heliocentric theory and just goes to prove how science can be improved on all the time and even changed completely.
