Announcement of Ṭūsī’s discovery of the rectilinear version of his couple, which was used later in the Ḥall for producing his non-Ptolemaic lunar and planetary models for motion in longitude. Since the Mu`īniyya can be dated to 1235 AD, that is the latest date for the discovery of this version of the Couple; dedicated to Mu`īn al-Dīn abū al-Shams, son of Nāṣir al-Dīn Muḥtasham, the Ismā`īlī governor of Qūhistān in Eastern Iran and Ṭūsī’s patron. There are 2 different versions of the introduction, one in which Ṭūsī praises his Ismā`īlī patrons, and the other which doesn't mention them (and thus written after the fall of Alamūt in 654/1256).   [Ragep]