Dia tōn grammōn: Hipparchus on simultaneous risings and settings

Abstract

In his Exegesis, Hipparchus explicitly states that he developed a method to calculate simultaneous risings and settings, It is generally assumed that Hipparchus used either Menelaus’ Theorem, or stereographic projections, even if both methods are only attested for later periods and their availability to Hipparchus is debated. In this article we argue that simultaneous risings and settings could be calculated using less advanced mathematical tools unquestionably available to Hipparchus: Elements I.47 (commonly known as the theorem of Pythagoras), the Rule of Three (applied to simple proportions between right triangles), and the Table of Chords. Without claiming that this was indeed the method that Hipparchus used, we argue that there is no need to postulate the availability of more sophisticated tools to carry out the calculations which Hipparchus mentions in the Exegesis.