The geocentric model, not only of the Solar System, but of the Universe held sway from the time of the ancient Greeks hundreds of years before Christ and was only challenged in the 16th century C.E. after which it was gradually replaced by the heliocentric view of the Solar System and a view of the Universe that has no particular center. The main proponents of the geocentric system were Aristotle and Ptolemy. Geocentrism was endorsed by the early Christians and the Catholic church because it appeared to be consistent with certain Biblical passages. (Joshua 10:12 in which the Sun and Moon are said to stop in the sky, and Psalms 93:1 in which the world is described as immobile.) Even today, some adherents of Biblical fundamentalism claim that the Earth is the center of the Universe and that the Sun revolves around the Earth.
The Ptolemaic geocentric system posited that the circle was a perfect shape and that all movements in the heavens were circular.
The latter was first challenged by the apparent movement of the planets in the sky with respect to the stars, particularly that of Mars, in its recurrent retrograde motion. It appears to move backward and form a loop for a brief time before resuming its regular path across the star field.
Hypatia’s fictional slave Davus makes a model that illustrates the solution proposed by Ptolemy to explain the retrograde motion of the planets preserving the circle as the only possible orbit (clip #2). Ptolemy’s explanation involves the superposition of two circular motions which, together, are able to explain the retrograde motion of Mars together with the other observed paths of the planets across the sky while keeping the Earth at its center.
But adding circle upon circle only made the models more complicated and it is an accepted principle among scientists to think that nature tends toward the simple. This idea is behind Occam’s Razor, a principle of logic by which the simpler option is preferred between competing theories, all else being equal. So, the Ptolemaic model with its epicycles seemed a strained explanation of the observed motion of the planets. To move away from it, however, it would take two more gigantic leaps in understanding that would have to wait more than a thousand years after the time of Hypatia.
The first new concept was that the Earth is not the center of the Universe and the second was to let go of the notion that the circle was the only possible orbit in the Universe.
Polish astronomer Nikolas Copernicus hypothesized in 1543 that the Sun, and not the Earth, was at the center of the Solar System. A debate followed for many years. Shortly afterward the Danish astronomer Tycho Brahe tried to come up with a compromise between both models, placing the Earth in the center of the universe and placing the Moon and the Sun in (circular) orbits around it. All the planets also had a circular orbit around the Sun. The spell of circularity was difficult to shake off.
The finding that orbits of the planets were not circular, but were instead elliptical was made by Johannes Kepler. For a summary of their discoveries see Nikolaus Copernicus, Tycho Brahe, & Johannes Kepler