Sharaf al-Din al-Tusi

Sharaf al-Din Muhammad ibn Masud al-Tusi was a distinguished Persian mathematician and astronomer born in 1135 CE in Tus, a historically significant city in the Khorasan region of present-day Iran. He lived during a period of great intellectual ferment in the Islamic world and devoted his life to the pursuit of mathematical knowledge, traveling widely and teaching in cities including Mosul and Damascus before returning to Persia.

Al-Tusi is best known for his groundbreaking work on cubic equations, which constitutes one of the most remarkable achievements in medieval Islamic mathematics. In his masterwork, the Mu'adalat (Treatise on Equations), he systematically studied cubic equations, classified them into numerous types, and developed methods for determining whether they had positive solutions and, if so, how many. His approach anticipated key concepts of calculus: he effectively found the maximum of a polynomial function in order to determine conditions under which a cubic equation has a solution, working with what amounts to a derivative condition centuries before the formal development of differential calculus.

He constructed numerical methods for approximating the roots of cubic equations to any desired degree of accuracy, work that was both more systematic and more sophisticated than anything his predecessors had achieved. His linear astrolabe, described in a separate treatise, was a simplification of the traditional astrolabe that made certain calculations more practical.

Al-Tusi taught mathematics at a high level and was known as an inspiring educator. He died around 1213 CE in Iran. His work on cubics was so advanced that its full significance was only recognized by modern historians of mathematics in the twentieth century, who identified his use of derivative-like reasoning as a profound anticipation of calculus.