Sharaf al-Dīn al-Ṭūsī | |
|---|---|
| Born | Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī c. 1135 Tus, present-day Iran |
| Died | c. 1213 |
| Occupation | Mathematician |
| Era | Islamic Golden Age |
Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī (Persian:شرفالدین مظفر بن محمد بن مظفر توسی;c. 1135Tus, Iran –c. 1213Iran)[1] known more often asSharaf al-Dīn al-Ṭūsī orSharaf ad-Dīn aṭ-Ṭūsī,[2] was anIranianmathematician andastronomer of theIslamic Golden Age (during theMiddle Ages).[3][4]
Al-Tusi was probably born inTus, Iran. Little is known about his life, except what is found in the biographies of other scientists[5] and that most mathematicians today can trace their lineage back to him.[6]
Around 1165, he moved toDamascus and taught mathematics there. He then lived inAleppo for three years, before moving toMosul, where he met his most famous discipleKamal al-Din ibn Yunus (1156-1242). Kamal al-Din would later become the teacher of another famous mathematician from Tus,Nasir al-Din al-Tusi.[5]
According toIbn Abi Usaibi'a, Sharaf al-Din was "outstanding ingeometry and the mathematical sciences, having no equal in his time".[7][a]
Al-Tusi has been credited with proposing the idea of a function, however his approach being not very explicit, algebra's decisive move to the dynamic function was made 5 centuries after him, by German polymath Gottfried Leibniz.[8]Sharaf al-Din used what would later be known as the "Ruffini-Horner method" tonumerically approximate theroot of acubic 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.[5] 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.[9][10][11] 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 thecoefficient 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).[b] For each of these five types, al-Tusi wrote down an expression m for the point where the function f(x) attained itsmaximum, 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.[12]
Al-Tusi gave no indication of how he discovered the expressions m for the maxima of the functions f(x).[13] 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.[14][15] 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.[16][17]
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 thediscriminant 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,[17]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.[18]
Sharaf al-Din analyzed the equationx3 +d =b⋅x2 in the formx2 ⋅ (b -x) =d, stating that the left hand side must at least equal the value ofd for the equation to have a solution. He then determined the maximum value of this expression. A value less thand means no positive solution; a value equal tod corresponds to one solution, while a value greater thand corresponds to two solutions. Sharaf al-Din's analysis of this equation was a notable development inIslamic mathematics, but his work was not pursued any further at that time, neither in the Muslim or European world.[19]
Sharaf al-Din al-Tusi's "Treatise on equations" has been described by Roshdi Rashed as inaugurating the beginning ofalgebraic geometry.[20] This was criticized by Jeffrey Oaks who claims that Al-Tusi did not study curves by means of equations, but rather equations by means of curves (just asal-Khayyam had done before him) and that the study of curves by means of equations originated with Descartes in the seventeenth century.[21][22]
Sharaf al-Din invented a linearastrolabe, sometimes called the "Staff of Tusi". While it was easier to construct and was known inal-Andalus, it did not gain much popularity.[7]
The main-belt asteroid7058 Al-Ṭūsī, discovered byHenry E. Holt atPalomar Observatory in 1990, was named in his honor.[23]
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