Ptolemy Through the Centuries

Summary Ptolemy’s theorem is a classical result obtained in the late Greek-Roman period, whose first application was to provide computational support to a geocentric cosmological model. This model’s most important achievement was that it explained the apparent movement of celestial bodies to a subjective observer on the Earth. What makes Ptolemy’s theorem a very interesting case in the history of mathematics is that the Euclidean concept of a Ptolemaic configuration can be investigated in the geometry of general metric spaces, in a situation very similar to the triangle inequality. To complement the historical narrative, in the final part of our paper we introduce a new norm, related to the Euclidean, Chebyshev, and Manhattan norms, and we investigate its properties in relation with other norms, hoping to illustrate how this fundamental configuration traversed Euclidean geometry, complex geometry, and analysis, transformational geometry, to become an interesting classification criterion in metric geometry.