'Abd al-Hamīd ibn Turk

ʿAbd al-Hamīd ibn Turk, also known as ʿAbd al-Hamīd ibn Wase ibn Turk al-Jili, was a ninth-century Islamic mathematician. Little is known about his life, with historical records from Ibn al-Nadim and Al-Qifti offering slightly different details about his name and origins. Al-Qifti suggests he was from Gilan, while Ibn Nadim's records point to him originating from Khuttal, a region north of the Oxus river. David Pingree, in the Encyclopaedia Iranica, also notes his possible origins from Khuttal or Gilan. His notable work is on algebra, titled *Logical Necessities in Mixed Equations*. This work is considered similar to Al-Khwarizmi's *Al-Jabr* and might have been published contemporaneously or even earlier. A surviving chapter focuses on solving quadratic equations, presenting geometric demonstrations that are identical to those in *Al-Jabr*. Interestingly, Ibn Turk's work goes further by providing a geometric proof for the case where the discriminant is negative, indicating no solution. The similarities between their works have led historians to suggest that algebra was a well-developed field by the time of both Al-Khwarizmi and 'Abd al-Hamid.