Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī (; 1135 – 1213) was an Iranian mathematician and astronomer of the Islamic Golden Age (during the Middle Ages). Biography Tusi was probably born in Tus, Iran. Little is known about his life, except what is found in the biographies of other scientists and that most mathematicians today can trace their lineage back to him. Around 1165, he moved to Damascus and taught mathematics there. He then lived in Aleppo for three years, before moving to Mosul, where he met his most famous disciple Kamal al-Din ibn Yunus (1156-1242). This Kamal al-Din would later become the teacher of another famous mathematician from Tus, Nasir al-Din al-Tusi. According to Ibn Abi Usaibi'a, Sharaf al-Din was "outstanding in geometry and the mathematical sciences, having no equal in his time". Mathematics Al-Tusi has been credited with proposing the idea of a function, however his approach being not very explicit, Algebra's move to the dynamic function was made 5 centuries after him, by Gottfried Leibniz. Sharaf al-Din used what would later be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He also developed a novel method for determining the conditions under which certain types of cubic equations would have two, one, or no solutions. The equations in question can be written, using modern notation, in the form , where is a cubic polynomial in which the coefficient of the cubic term is , and is positive. The Muslim mathematicians of the time divided the potentially solvable cases of these equations into five different types, determined by the signs of the other coefficients of . For each of these five types, al-Tusi wrote down an expression for the point where the function attained its maximum, and gave a geometric proof that for any positive different from . He then concluded that the equation would have two solutions if , one solution if , or none if . Al-Tusi gave no indication of how he discovered the expressions for the maxima of the functions . Some scholars have concluded that al-Tusi obtained his expressions for these maxima by "systematically" taking the derivative of the function , and setting it equal to zero. This conclusion has been challenged, however, by others, who point out that al-Tusi nowhere wrote down an expression for the derivative, and suggest other plausible methods by which he could