Sharaf al-Din al-Tusi

Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī (Persian: شرف‌الدین مظفر بن محمد بن مظفر توسی; c. 1135 Tus, Iran – c. 1213 Iran)^[1] known more often as Sharaf al-Dīn al-Ṭūsī or Sharaf ad-Dīn aṭ-Ṭūsī,^[2] was an Iranian mathematician and astronomer of the Islamic Golden Age (during the Middle Ages).^[3][4]

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

For other people with similar names, see Al-Tusi.

Biography

Al-Tusi was probably born in Tus, 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 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). Kamal al-Din would later become the teacher of another famous mathematician from Tus, Nasir al-Din al-Tusi.^[5]

According to Ibn Abi Usaibi'a, Sharaf al-Din was "outstanding in geometry and the mathematical sciences, having no equal in his time".^[7][a]

Mathematics

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" 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.^[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 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).^[b] 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 .^[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 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 ,^[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 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 or European world.^[19]

Sharaf al-Din al-Tusi's "Treatise on equations" has been described by Roshdi Rashed as inaugurating the beginning of algebraic 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 as al-Khayyam had done before him) and that the study of curves by means of equations originated with Descartes in the seventeenth century.^[21][22]

Astronomy

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.^[7]

Honours

The main-belt asteroid 7058 Al-Ṭūsī, discovered by Henry E. Holt at Palomar Observatory in 1990, was named in his honor.^[23]

Notes

1. Mentioned in the biography of the Damascene architect and physician Abu al-Fadhl al-Harithi (d. 1202-3).^{[citation needed]}
2. The five types were:
   1. a x2 − x3 = c
   2. b x − x3 = c
   3. b x − a x2 − x3 = c
   4. −b x + a x2 − x3 = c
   5. b x + a x2 − x3 = c
   where  a  and  b  are positive numbers.^[9] For any other values of the coefficients of  x  and  x2, the equation  f(x) = c  has no positive solution.

1. Brummelen, Glen van (2007). "Sharaf al‐Dīn al‐Ṭūsī". In Hockey, Thomas; et al. (eds.). Biographical Encyclopedia of Astronomers. New York: Springer. p. 1051. doi:10.1007/978-0-387-30400-7_1268. ISBN 978-0-387-31022-0. Retrieved 2023-06-18.
2. "Sharaf ad-Dīn aṭ-Ṭūsī". zbMATH Open (Author Profile). Retrieved 2023-06-18.
3. 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'"
4. Nasehpour 2018.
5. O'Connor & Robertson 1999.
6. Mathematics Genealogy Project Extrema
7. Berggren 2008.
8. Nasehpour 2018, "apparently the idea of a function was proposed by the Persian mathematician Sharaf al-Din al-Tusi (died 1213/4), though his approach was not very explicit, perhaps because of this point that dealing with functions without symbols is very difficult. Anyhow algebra did not decisively move to the dynamic function substage until the German mathematician Gottfried Leibniz(1646–1716)."
9. Hogendijk 1989, p. 71.
10. Hogendijk 1997, p. 894.
11. Smith 1997b, p. 69.
12. Hogendijk 1989, pp. 71–72.
13. Berggren 1990, pp. 307–308.
14. Rashed 1994, p. 49.
15. Farès 1995.
16. Berggren 1990.
17. Hogendijk 1989.
18. Rashed 1994, pp. 46–47, 342–43.
19. Katz, Victor; Barton, Bill (October 2007). "Stages in the History of Algebra with Implications for Teaching". Educational Studies in Mathematics. 66 (2): 192. doi:10.1007/s10649-006-9023-7. S2CID 120363574.
20. Rashed 1994, pp. 102-3.
21. Brentjes, Sonja; Edis, Taner; Richter-Bernburg, Lutz (2016). 1001 Distortions: How (Not) to Narrate History of Science, Medicine, and Technology in Non-Western Cultures. Ergon Verlag. p. 158.
22. Oaks, Jeffrey (2016). "Excavating the errors in the "Mathematics" chapter of 1001 Inventions". Academia.edu.
23. "7058 Al-Tusi (1990 SN1)". Minor Planet Center. Retrieved 21 November 2016.

References

• 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 – via Encyclopedia.com.
• Farès, Nicolas (1995), "Le calcul du maximum et la 'dérivée' selon Sharaf al-Din al-Tusi" (PDF), Arabic Sciences and Philosophy, 5 (2): 219–317, doi:10.1017/s0957423900002034, S2CID 170242949
• 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
• Nasehpour, Peyman (August 2018). "A Brief History of Algebra with a Focus on the Distributive Law and Semiring Theory". arXiv:1807.11704. Bibcode:2018arXiv180711704N. S2CID 119176936. ResearchGate:326732377.
• 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
• 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
• 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
• Smith, Julian A. (1997a), "Astrolabe", in Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, pp. 74–75, ISBN 9780792340669

Further reading

• Anbouba, Adel (2008). "Al-Ṭūsī, Sharaf Al-dīn Al-Muẓaffar Ibn Muḥammad Ibn Al-Muẓaffar". Complete Dictionary of Scientific Biography. Vol. 13. Charles Scribner's Sons. pp. 514–517. Gale CX2830904401.