Sharaf al-Din al-Tusi

Tusi was probably born in Tus, Iran. Little is known about his life, except what is found in the biographies of other scientists[3] and that most mathematicians today can trace their lineage back to him.[4]

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.[3]

According to Ibn Abi Usaibi'a, Sharaf al-Din was "outstanding in geometry and the mathematical sciences, having no equal in his time".[5][6]

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.[7] 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.[8] 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).[9] 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 .[10]

Al-Tusi gave no indication of how he discovered the expressions  m  for the maxima of the functions  f(x).[11] 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.[12] 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.[13]

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 ,[14] 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.[15]

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. 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.[16]

Sharaf al-Din al-Tusi's "Treatise on equations" has been described as inaugurating the beginning of algebraic geometry.[17]

Sharaf al-Din invented a linear astrolabe, sometimes called the "staff of Tusi". While it was easier to construct and was known in al-Andalus, it did not gain much popularity.[5]

The main-belt asteroid 7058 Al-Ṭūsī, discovered by Henry E. Holt at Palomar Observatory in 1990, was named in his honor.[18]

• O'Connor, John J.; Robertson, Edmund F. (1999), "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics archive, University of St Andrews.
• Berggren, J. Lennart (1990), "Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's Muʿādalāt", Journal of the American Oriental Society, 110 (2): 304–309, doi:10.2307/604533, JSTOR 604533
• Berggren, J. Lennart (2008). "Al-Tūsī, Sharaf Al-Dīn Al-Muzaffar Ibn Muhammad Ibn Al-Muzaffar". Complete Dictionary of Scientific Biography. Charles Scribner & Sons. Retrieved March 21, 2011 from Encyclopedia.com.
• Hogendijk, Jan P. (1989), "Sharaf al-Dīn al-Ṭūsī on the Number of Positive Roots of Cubic Equations", Historia Mathematica, 16: 69–85, doi:10.1016/0315-0860(89)90099-2
• Farès, Nicolas (1995), "Le calcul du maximum et la 'dérivée' selon Sharaf al-Din al-Tusi", Arabic Sciences and Philosophy, 5 (2): 219–317, doi:10.1017/s0957423900002034
• Hogendijk, Jan P. (1997), "Sharaf al-Dīn al-Ṭūsī", in Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, p. 894, ISBN 9780792340669
• Rashed, Roshdi (1994), The Development Of Arabic Mathematics: Between Arithmetic And Algebra, translated by Armstrong, A.F.W., Dordrecht: Springer Science+Business Media, ISBN 978-90-481-4338-2
• Selin, Helaine, ed. (1997), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (1st ed.), Dordrecht: Kluwer Academic Publishers, ISBN 0-7923-4066-3
• Smith, Julian A. (1997a), "Astrolabe", in Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, pp. 74–75, ISBN 9780792340669
• Smith, Julian A. (1997b), "Arithmetic in Islamic Mathematics", in Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, pp. 68–70, ISBN 9780792340669

• Brummelen, Glen van (2007). "Sharaf al‐Dīn al‐Ṭūsī". In Thomas Hockey; et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. p. 1051. ISBN 978-0-387-31022-0. (PDF version)