Sharaf al-Dīn al-Ṭūsī
|Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī|
Tus, present-day Iran
|Era||Islamic Golden Age|
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.
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.
- Smith (1997a, p.75),"This was invented by Iranian mathematician Sharaf al-Din al-Tusi (d. ca. 1213), and was known as "Al-Tusi's cane""
- O'Connor & Robertson (1999).
- Berggren 2008.
- 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).
- The five types were:
- a x2 − x3 = c
- b x − x3 = c
- b x − a x2 − x3 = c
- −b x + a x2 − x3 = c
- b x + a x2 − x3 = c
- Hogendijk (1989, p.71–2).
- Berggren (1990, p.307–8).
- Rashed (1994, p.49), Farès (1995).
- Berggren (1990), Hogendijk (1989).
- Hogendijk (1989).
- Rashed (1994, pp.46–47, 342–43).
- Rashed (1994, pp.102-3)
- "7058 Al-Tusi (1990 SN1)". Minor Planet Center. Retrieved 21 November 2016.
- 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
- Farès, Nicolas (1995), "Le calcul du maximum et la 'dérivée' selon Sharaf al-Din al-Tusi", Arabic Sciences and Philosophy, 5 (2): 219–317
- 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
- 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
- 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