Dia tōn grammōn : Hipparchus on simultaneous risings and settings.

This article focuses on reconstructing a method Hipparchus, the Hellenistic astronomer, might have used to calculate simultaneous risings and settings of stars and constellations, as described in his Exegesis. It argues that Hipparchus could have solved this problem using only basic mathematical tools available in his time—namely, Euclid's Elements I.47 (the Pythagorean theorem), the Rule of Three for proportions, and a Table of Chords—without requiring more advanced techniques like Menelaus' Theorem, stereographic projections, or the analemma, whose availability to Hipparchus is uncertain. The proposed geometrical procedure involves breaking down the problem into three steps: determining the culminating point of the celestial equator, converting this to the ecliptic coordinate system, and then finding the rising point of the ecliptic, all through plane geometry and proportional reasoning. While computationally intensive, this method demonstrates that Hipparchus' calculations could have been achieved with simpler tools, though the article does not claim this was his actual method, acknowledging that he may have relied on a celestial globe or other means.