This article investigates the origin of the Sector Theorem, generally known as the Theorem of Menelaus. Building on earlier work by R. Lorch [On the sector-figure and related texts. Edited with translation, preface and commentary by Richard Lorch. Frankfurt am Main: Institute for the History of Arabic-Islamic Science (2001; Zbl 1013.01014)], Krause, and Björnbo, the author first considers the Greek, Arabic and Latin history of the text of the theorem, in which he distinguishes three traditions: astronomical (exemplified by Ptolemy’s Almagest), geometrical (Menelaus’s Spherics), and didactic (Theon of Alexandria’s Commentary on the Almagest and Thābit ibn Qurra’s On the Sector Theorem). He then gives an overview of all known (but in many cases non-extant) versions of the Spherics of Menelaus, and identifies al-Harawī’s Arabic revision of al-Māhānī’s partial reworking and Gerard of Cremona’s more complete Latin translation based on Ibn Ḥunayn as the closest witnesses to Menelaus’s original Sector Theorem. Of both sources he includes the original text (in Appendix A) and an English translation (pp. 53 and 55–57), supplemented with a mathematical commentary in Appendix B.
By now looking in detail at the role of the Sector Theorem in the spherical trigonometry of Menelaus’s Spherics and in the calculation of rising times in Ptolemy’s Almagest, and by taking into account the only very few ancient references to Hipparchus’s treatment of spherical astronomical problems, the author concludes that Menelaus was very probably not the original author of the Sector Theorem, but adapted it for his purposes from an established theorem of quantative spherical astronomy that had already been in use at the time of Hipparchus.