**Sharaf al-Tusi** was an Islamic mathematician who wrote a treatise on cubic equations.

- 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.
- 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.
- 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.
- We do have a number of works by al-Tusi which are important in the development of mathematics.
- What is in this Treatise on equations by al-Tusi?
- First al-Tusi discusses twelve types of equation of degree at most two.
- The method which al-Tusi used is quite remarkable.
- We use, of course, modern notation to make the solution easy to understand, while al-Tusi would express all his mathematics in words.
- Of course al-Tusi's use of the derivative of a function, without of course saying so, is very interesting.
- 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 $n$th 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.

Born about 1135, Tus, Khorasan (now Iran). Died 1213, Iran.

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Ancient Arab, Astronomy, Origin Iran

Epochs: 1

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive