Recovering Two Lost Treatises on Approximating the Sine of 1 ° from Qushjī’s Commentary on Zīj-i Ṣulṭānī

The accuracy of medieval approximations of the Sine of 1° reached its peak with the works of the Persian mathematician and astronomer Ghiyāth al-Dīn Jamshīd al-Kāshī (d. 832/1429), and his patron Ulugh Beg (d. 853/1449), the Timurid ruler of Transoxiana and a mathematician and astronomer in his own right. Their works were written during the active phase of the Samarkand observatory, which was founded by Ulugh Beg and whose final product, Zīj-i Sulṭānī, was the most accurate zīj of the medieval period. Even though neither of their treatises on approximating the Sine of 1° has reached us, Kāshī’s and Ulugh Beg’s approximation methods were transmitted through the works of their colleagues at the observatory, namely Qāḍī-zāda al-Rūmī’s (d. after 844/1441) recension of Kāshī’s treatise and ʿAlāʾ al-Dīn ʿAlī al-Qūshjī’s (d. 879/1474) commentary on Zīj-i Sulṭānī. Unlike Qāḍī-zāda’s treatise, Qūshjī’s commentary has not received the attention it deserves from historians. Thus, it has not been noticed that what is presented in Qūshjī’s commentary under the rubric of “Ulugh Beg’s demonstrative method” is, in fact, a synthesis of Kāshī’s and Ulugh Beg’s approximation methods. The present article aims to fill this gap by offering an edition and English translation of the relevant passages of Qūshjī’s commentary, disti