# 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.

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.

According to Ibn Abi Usaibi’a, Sharaf al-Din was “outstanding in geometry and the mathematical sciences, having no equal in his time”.

## Mathematics

Al-Tusi has been credited with proposing the idea of a function, however his approach being not explicit enough, Algebra’s move to the dynamic function was made 5 centuries after him, by Gottfried Leibniz. 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.

Sharaf al-Din analyzed the equation x3 + d = bx2 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.

Sharaf al-Din al-Tusi’s “Treatise on equations” has been described as inaugurating the beginning of algebraic geometry.

## Astronomy

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

## Honours

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

Born  Ṭūs, (Iran)circa 1135

Died  (Iran), 1213

Although Sharaf al‐Dīn al‐Ṭūsī is known especially for his mathematics (in particular his novel work on the solutions of cubic equations), he was also the inventor of the linear astrolabe, a tool that derives from the planispheric astrolabe but is more easily constructed. From his name we may infer that Sharaf al‐Dīn was born in the region of Ṭūs, in northeastern Iran. He spent a major part of his early career as a teacher of the sciences, including astronomy and astrology, in Damascus and Aleppo; he also taught in Mosul. Among his students was Kamāl al‐Dīn ibn Yūnus, who would eventually teach Sharaf’s namesake, the great Naṣīr al‐Dīn al‐Ṭūsī.

Sharaf al‐Dīn al‐Ṭūsī devoted several treatises to the linear astrolabe, sometimes called the staff of al‐Ṭūsī. Its principle is simple – many of the important circles on the planispheric astrolabe, especially the almucantars (altitude circles) and the circles of declination, are centered on the meridian line. The main rod of the linear astrolabe is equivalent to the meridian line and contains markings to indicate the centers of these circles and their intersections with the meridian. The ecliptic (which appears on the movable rete of a standard astrolabe) is represented by the intersections of the beginnings of the zodiacal signs with the meridian when the rete is rotated. Many typical operations on a traditional astrolabe require the locations of points of intersection of these various circles. By attaching ropes to the appropriate points on the staff to act as radii, the circles and their intersections can be reconstructed and the astronomical problem solved. A scale giving chord lengths in the meridian circle extended the linear astrolabe’s range of applications. Attached to a plumb line, it was also used to take observations of solar altitude. Additional markings allowed the determination of the qibla (the direction of Mecca) and solutions of astrological problems.

The simplicity of the linear astrolabe made it easy to construct, but its less than artful appearance rendered it unattractive to collectors. It was neither as durable nor as accurate as a planispheric astrolabe, and its operations were less intuitive. None have survived.