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Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī (شرف‌الدین طوسی) (1135 - 1213) was a Persian mathematician and astronomer of the Islamic Golden Age (during the Middle Ages).

## Biography

Tusi taught various mathematical topics including the science of numbers, astronomical tables and astrology, in Aleppo and Mosul. His best pupil was Kamal al-Din ibn Yunus. In turn Kamal al-Din ibn Yunus went on to teach Nasir al-Din al-Tusi, one of the most famous of all the Islamic scholars of the period. By this time Tusi seems to have acquired an outstanding reputation as a teacher of mathematics, for some travelled long distances hoping to become his students.

## Works

Al-Tusi wrote some treatises on algebra. There, he went on to give what we would essentially call the Ruffini-Horner method for approximating the root of a cubic equation. Although this method had been used by earlier Arabic mathematicians to find approximations for the nth root of an integer, Tusi is the first that we know who applied the method to solve general equations of this type.[1]

### Treatise on Equations

In his Al-Mu'adalat (Treatise on Equations), al-Tusi found algebraic and numerical solutions of cubic equations and was the first to discover the derivative of cubic polynomials, an important result in differential calculus.[2]

Al-Tusi's Treatise on Equations dealt with equations up to the third degree. The treatise does not follow Al-Karaji's school of algebra, but instead represents "an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry." The treatise dealt with 25 types of equations, including twelve types of linear equations and quadratic equations, eight types of cubic equations with positive solutions, and five types of cubic equations which may not have positive solutions. One of the most remarkable aspects of this work is the development of concepts related to calculus, such as the derivative function and the maxima and minima of curves, in order to solve cubic equations which may not have positive solutions. For example, in order to solve the equation $\ x^3 + a = bx$, al-Tusi finds the maximum point of the curve $\ bx - x^3 = a$. He uses the derivative of the function to find that the maximum point occurs at $x = \sqrt{\frac{b}{3}}$, and then finds the maximum value for y at $2(\frac{b}{3})^\frac{3}{2}$ by substituting $x = \sqrt{\frac{b}{3}}$ back into $\ y = bx - x^3$. He finds that the equation $\ bx - x^3 = a$ has a solution if $a \le 2(\frac{b}{3})^\frac{3}{2}$, and al-Tusi thus deduces that the equation has a positive root if $D = \frac{b^3}{27} - \frac{a^2}{4} \ge 0$, where $D$ is the discriminant of the equation.[1] He understood the importance of the discriminant of the cubic equation and used an early version of Cardano's formula[3] to find algebraic solutions to certain types of cubic equations.[2]

Sharaf al-Din also developed the concept of a function. In his analysis of the equation $\ x^3 + d = bx^2$ for example, he begins by changing the equation's form to $\ x^2 (b - x) = d$. He then states that the question of whether the equation has a solution depends on whether or not the “function” on the left side reaches the value $\ d$. To determine this, he finds a maximum value for the function. He proves that the maximum value occurs when $x = \frac{2b}{3}$, which gives the functional value $\frac{4b^3}{27}$. Sharaf al-Din then states that if this value is less than $\ d$, there are no positive solutions; if it is equal to $\ d$, then there is one solution at $x = \frac{2b}{3}$; and if it is greater than $\ d$, then there are two solutions, one between $\ 0$ and $\frac{2b}{3}$ and one between $\frac{2b}{3}$ and $\ b$.[4]

### Numerical analysis

In numerical analysis, the essence of Viète's method was known to al-Tusi, and it is possible that the algebraic tradition of al-Tusi, as well as his predecessor Omar Khayyám and successor Jamshīd al-Kāshī, was known to 16th century European algebraists, or whom François Viète was the most important.[5]

A method algebraically equivalent to Newton's method was also known to al-Tusi. His successor al-Kāshī later used a form of Newton's method to solve $x^P - N = 0$ to find roots of N. In western Europe, a similar method was later described by Henry Biggs in his Trigonometria Britannica, published in 1633.[6]

### Linear astrolabe

Another famous work by Tusi is one in which he describes the linear astrolabe, sometimes called the "staff of al-Tusi", which he invented. It was "a simple wooden rod with graduated markings but without sights. It was furnished with a plumb line and a double chord for making angular measurements and bore a perforated pointer."[1]

## Notes

1. 1.0 1.1 1.2
2. 2.0 2.1 J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2), p. 304-309.
3. Rashed, Roshdi; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 342–3, ISBN 0792325656
4. Victor J. Katz, Bill Barton (October 2007), "Stages in the History of Algebra with Implications for Teaching", Educational Studies in Mathematics (Springer Netherlands) 66 (2): 185–201 [192], doi:10.1007/s10649-006-9023-7
5. Ypma, Tjalling J. (December 1995), "Historical Development of the Newton-Raphson Method", SIAM Review (Society for Industrial and Applied Mathematics) 37 (4): 531–551 [534]
6. Ypma, Tjalling J. (December 1995), "Historical Development of the Newton-Raphson Method", SIAM Review (Society for Industrial and Applied Mathematics) 37 (4): 531–551 [539]