Ghyath al-Din Jamshid Kashani

 At the time that Kashani was growing up Timur (often known as Tamburlaine) was conquering large regions. He had proclaimed himself sovereign and restorer of the Mongol empire at Samarkand in 1370 and, in 1383, Timur began his conquests in Persia with the capture of Herat. Timur died in 1405 and his empire was divided between his two sons, one of whom was Shah Rokh.

While Timur was undertaking his military campaigns, conditions were very difficult with widespread poverty. Kashani lived in poverty, like so many others at this time, and devoted himself to astronomy and mathematics while moving from town to town. Conditions improved markedly when Shah Rokh took over after his father's death. He brought economic prosperity to the region and strongly supported artistic and intellectual life. With the changing atmosphere, Kashani's life also improved. The first event in Kashani's life which we can date accurately is his observation of an eclipse of the moon which he made in Kashan on 2 June 1406 (He dated many of his works with the exact date on which they were completed.).

It is reasonable to assume that Kashani remained in Kashan where he worked on astronomical texts. He was certainly in his home town on 1 March 1407 when he completed Sullam Al-sama the text of which has survived. The full title of the work means The Stairway of Heaven, on Resolution of Difficulties Met by Predecessors in the Determination of Distances and Sizes (of the heavenly bodies). At this time it was necessary for scientists to obtain patronage from their kings, princes or rulers. Kashani played this card to his advantage and brought himself into favour in the new era where patronage of the arts and sciences became popular. His Compendium of the Science of Astronomy written during 1410-11 was dedicated to one of the descendants of the ruling Timurid dynasty.

Samarkand, in Uzbekistan, is one of the oldest cities of Central Asia. The city became the capital of Timur's empire and Shah Rokh and his son, Ulugh Beg, ruler of the city. Ulugh Beg, himself a great scientist, began to build the city into a great cultural centre. It was to Ulugh Beg that Kashani dedicated his important book of astronomical tables Khaqani Zij which was based on the tables of Nasir al-Tusi. In the introduction Kashani says that without the support of Ulugh Beg he could not have been able to complete it. In this work there are trigonometric tables giving values of the sine function to four sexagesimal digits for each degree of argument with differences to be added for each minute. There are also tables which give transformations between different coordinate systems on the celestial sphere, in particular allowing ecliptic coordinates to be transformed into equatorial coordinates.

Kashani had certainly found the right patron in Ulugh Beg since he founded a university for the study of theology and science at Samarkand in about 1420 and he sought out the best scientists to help with his project. Ulugh Beg invited Kashani to join him at this school of learning in Samarkand, as well as around sixty other scientists including Qazi Zadeh. There is little doubt that Kashani was the leading astronomer and mathematician at Samarkand and he was called the second Ptolemy by an historian writing later in the same century.

In 1424 Ulugh Beg began the construction of an observatory in Samarkand and, although the letters by Kashani are undated they were written at a time when construction of the observatory had begun. The contents of one of these letters has only recently been published.

In the letters Kashani praises the mathematical abilities of Ulugh Beg but of the other scientists in Samarkand, only Qazi Zadeh earned his respect. Ulugh Beg led scientific meetings where problems in astronomy were freely discussed. Usually these problems were too difficult for all except Kashani and Qazi Zadeh and on a couple of occasions only Kashani succeeded. It is clear that Kashani was the best scientist and closest collaborator of Ulugh Beg at Samarkand and, despite Kashani's ignorance of the correct court behaviour and lack of polished manners, he was highly respected by Ulugh Beg. After Kashani's death, Ulugh Beg described him as:

... a remarkable scientist, one of the most famous in the world, who had a perfect command of the science of the ancients, who contributed to its development, and who could solve the most difficult problems.

Although Kashani had done some fine work before joining Ulugh Beg at Samarkand, his best work was done while in that city. He produced his Treatise on the Circumference in July 1424, a work in which he calculated 2p to nine sexagesimal places and translated this into sixteen decimal places. This was an achievement far beyond anything which had been obtained before, either by the ancient Greeks or by the Chinese (who achieved 6 decimal places in the 5th century). It would be almost 200 years before van Ceulen surpassed Kashani's accuracy with 20 decimal places.

Kashani's most impressive mathematical work was, however, The Key to Arithmetic which he completed on 2 March 1427. The work is a major text intended to be used in teaching students in Samarkand, in particular Kashani tries to give the necessary mathematics for those studying astronomy, surveying, architecture, accounting and trading. The authors of Dictionary of Scientific Biography (New York 1970-1990) describe the work as follows:

In the richness of its contents and in the application of arithmetical and algebraic methods to the solution of various problems, including several geometric ones, and in the clarity and elegance of exposition, this voluminous textbook is one of the best in the whole of medieval literature; it attests to both the author's erudition and his pedagogical ability.

 In this work Kashani computed sin 1 to the same accuracy as he had computed p in his earlier work. He also considered the equation associated with the problem of trisecting an angle, namely a cubic equation. He was not the first to look at approximate solutions to this equation since Abu Raihan Biruni had worked on it earlier. However, the iterative method proposed by Kashani is described by Dictionary of Scientific Biography:

... one of the best achievements in medieval algebra. ... But all these discoveries of al-Kashi's were long unknown in Europe and were studied only in the nineteenth and twentieth centuries by ... historians of science....