Tusi was probably born in Tus, Iran. Little is known about his life, except what is found in the biographies of other scientists.
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".
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 f(x) = c, where f(x) is a cubic polynomial in which the coefficient of the cubic term x3 is −1, and c 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 f(x). For each of these five types, al-Tusi wrote down an expression m for the point where the function f(x) attained its maximum, and gave a geometric proof that f(x) < f(m) for any positive x different from m. He then concluded that the equation would have two solutions if c < f(m), one solution if c = f(m), or none if f(m) < c.
Al-Tusi gave no indication of how he discovered the expressions m for the maxima of the functions f(x). Some scholars have concluded that al-Tusi obtained his expressions for these maxima by "systematically" taking the derivative of the function f(x), 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 have discovered his expressions for the maxima.
The quantities D = f(m) − c which can be obtained from al-Tusi's conditions for the numbers of roots of cubic equations by subtracting one side of these conditions from the other is today called the discriminant of the cubic polynomials obtained by subtracting one side of the corresponding cubic equations from the other. Although al-Tusi always writes these conditions in the forms c < f(m), c = f(m), or f(m) < c, rather than the corresponding forms D > 0 , D = 0 , or D < 0 ,Roshdi Rashed nevertheless considers that his discovery of these conditions demonstrated an understanding of the importance of the discriminant for investigating the solutions of cubic equations.
Sharaf al-Din analyzed the equation x3 + d = b⋅x2 in the form x2 ⋅ (b - x) = d, stating that the left hand side must at least equal the value of d for the equation to have a solution. He then determined the maximum value of this expression. It is arguable that the isolation of this expression is an early approach to the notion of a "function". A value less than d means no positive solution; a value equal to d corresponds to one solution, while a value greater than d corresponds to two solutions. Sharaf al-Din's analysis of this equation was a notable development in Islamic mathematics, but his work was not pursued any further at that time, neither in the Muslim world nor in Europe.
Sharaf al-Din al-Tusi's "Treatise on equations" has been described as inaugurating the beginning of algebraic geometry.
Sharaf al-Din invented a linear astrolabe, sometimes called the "staff of Tusi". While easier to construct and was known in al-Andalus, it did not gain much popularity.
^Mentioned in the biography of the Damascene architect and physician Abu al-Fadhl al-Harithi (d. 1202-3).
^O'Connor & Robertson (1999). To al-Tusi, "solution" meant "positive solution", since the possibility of zero or negative numbers being considered genuine solutions had yet to be recognised at the time (Hogendijk, 1989, p.71; 1997, p.894; Smith, 1997b, p.69).
Brummelen, Glen van (2007). "Sharaf al‐Dīn al‐Ṭūsī". In Thomas Hockey; et al. The Biographical Encyclopedia of Astronomers. New York: Springer. p. 1051. ISBN 978-0-387-31022-0. (PDF version)
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