**Sharaf al-Din al-Tusi**'s full name is Sharaf al-Din Al-Muzaffar ibn Muhammad ibn Al-Muzaffar al-Tusi. Very little is known of his life but a few details can be reconstructed from references that occur in works about other scientists of the time.

We can certainly deduce from his name that he was born in the region of Tus. This region, in northeastern Iran, includes the towns of Tus and the close-by town of Meshed, both high up in the valley of the Kashaf River. Nishapur, which is 75 km west of Tus, is in the same region and it would be impossible without discovering further information to be precise about which town of the Tus region that he was born in.

What is certain is that he spent a large part of his life teaching in different towns over quite a wide area. The Seljuq Turks had captured Damascus in Syria in 1154 and made it the capital of their large empire. The city prospered and many, including al-Tusi, were attracted to it. Certainly around 1165 Al-Tusi was in Damascus for there he taught Abu'l Fadl about the works of Euclid and Ptolemy. Abu'l Fadl was an interesting person, for he had started out as a carpenter before studying mathematics with al-Tusi. From Damascus it would appear that Al-Tusi remained in Syria, going from the largest to the second largest city of Syria, namely Aleppo.

Al-Tusi must have taught in Aleppo for at least three years, and it is interesting that there he taught an important member of the Jewish community of that city. Aleppo contained both a Jewish and Muslim community and around 50 years earlier it had played a major part in the Muslim resistance to the crusaders, who had unsuccessfully besieged the city. In Aleppo al-Tusi taught various mathematical topics including the science of numbers, astronomical tables and astrology.

From Aleppo, al-Tusi must have gone to Mosul, a city in northwestern Iraq, situated on the right bank of the Tigris River. At this time the city was at its political high point, being under the rule of the Zangid dynasty. In Mosul Al-Tusi taught his most famous pupil 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 al-Tusi seems to have acquired an outstanding reputation as a teacher of mathematics for some travelled long distances hoping to become his students.

Al-Tusi was probably still in Mosul when Saladin (who had himself been brought up in Mosul) moved his forces into Syria to begin his policy of uniting, partly by force and partly by diplomacy, the area of Syria, Mesopotamia, Palestine, and Egypt. Saladin captured Damascus in 1174 and at about this time al-Tusi left Mosul and returned to Iran. He taught in Baghdad towards the end of his life and it was during this period that he wrote his famous work on algebra which we shall describe below.

We do have a number of works by al-Tusi which are important in the development of mathematics. The most important is described by Sarton [5] as:-

... a treatise on algebra ...[written]in1209 [which]is only known through a commentary by an unknown author.

Sarton's use of the word "commentary" is a little misleading, since the unknown author of the manuscript writes (see for example [4]):-

In this work I wanted to summarise the art of algebra and al-muqabala, adapt what has survived from the great philosopher Sharaf al-Din al-Muzaffar ibn al-Muzaffar ibn Muhammad al-Tusi, and reduce his over lengthy exposition to a moderate size; I eliminated the tables he drew up to make his computations and solve his problems.

What is in this *Treatise on equations* by al-Tusi? Basically it is a treatise on cubic equations, but it does not follow the general development that came through al-Karaji's school of algebra. Rather, as Rashed writes in [2]:-

... it represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.

In the treatise equations of degree at most three are divided into 25 different types. First al-Tusi discusses twelve types of equation of degree at most two. He then looks at eight types of cubic equation which always have a positive solution, then five types which may have no positive solution.

The method which al-Tusi used is quite remarkable. We illustrate the method by showing how al-Tusi examined one of the five types of equation which under certain conditions has a solution, namely the equation *x*^{3} + *a* = *bx*, where *a*, *b* are positive. We use, of course, modern notation to make the solution easy to understand, while al-Tusi would express all his mathematics in words. Al-Tusi's first comment is that if *t* is a solution to this equation then *t*^{3} + *a* = *bt* and, since *a* > 0, *t*^{3} < *bt* so *t* < √*b*.

Next al-Tusi notes that *bx* - *x*^{3} = *a* and he then finds where the maximum of *y* = *bx* - *x*^{3} occurs. Basically using the derivative of this expression, al-Tusi finds the maximum occurs at *x* = √(*b*/3) and then finds the maximum value for *y* of 2(*b*/3)^{3/2} by substituting *x* = √(*b*/3) back into *y* = *bx* - *x*^{3}. Thus the equation *bx* - *x*^{3} = *a* has a solution if *a* ≤ 2(*b*/3)^{3/2}. Then Al-Tusi deduces that the equation has a positive root if

D=b^{3}/27 -a^{2}/4 ≥ 0

where *D* is the discriminant of the equation.

Of course al-Tusi's use of the derivative of a function, without of course saying so, is very interesting. The paper [11] attempts to discover the origin of this implicit use of the derivative, which the author claims arises from algebraic proofs based on analytical procedures. The paper [12] suggests that a rather different approach, not one analogous to the modern derivative, lay behind Al-Tusi's method. The papers [10] and [14] contribute to this discussion; see also [2], [3] and [4] for further insights.

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

Another famous work by al-Tusi is one in which he describes the linear astrolabe, sometimes called the "staff of al-Tusi", which he invented. It was [1]:-

... 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.

**Article by:** *J J O'Connor* and *E F Robertson*

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