The Islamic mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), in his "Treatise on Equations", established conditions for some cubic equations to have solutions, by finding the maxima of appropriate cubic polynomials. He proved, for example, that the maximum of the cubic occurs when , and concluded therefrom that the equation has exactly one positive solution when , and two positive solutions whenever . The historian of science, Roshdi Rashed, has argued that al-Tūsī must have used the derivative of the cubic to obtain this result. Rashed's conclusion has been contested by other scholars, however, who argue that he could have obtained the result by other methods which do not require the derivative of the function to be known.