Discussing Ptolemy's model and its equivalence with the Copernican model

Abstract We reviewed Ptolemy's geocentric model and showed step by step how to determine the longitude of a planet using a modern mathematical approach to facilitate its understanding. We also discussed Ptolemy's considerations for fixing the planets' latitude value. A geometric equivalence between the Ptolemy and Copernicus models are presented, highlighting the restrictions with which this equivalence was built. Furthermore, the aspects associated with the transition between Ptolemy's geocentric model and Copernicus's heliocentric model are discussed. We considered aspects about the use of epicycles over epicycles, as well as the question of the precision of the Ptolemy model in relation to the Copernicus model. In general, in this work we present a quantitative and qualitative view of Ptolemy's model so that the reader has a simple mathematical formulation available to determine the position of the planets. Additionally, to understand how Copernicus' proposal, even though it is geometrically equivalent to Ptolemy's model, turned out to be a physically more complete model and of a different aesthetic (movement of the Earth around the Sun) capable of unifying concepts.