Ghyath al-Din Jamshid Kashani was a prominent 15th century Persian astronomer and mathematician who lived during a time of political turmoil in Persia under Timur and Shah Rokh. He lived in poverty as a young man but gained patronage from Ulugh Beg, the ruler of Samarkand, who invited Kashani to join his new university. There, Kashani flourished as the leading scientist and closely collaborated with Ulugh Beg. Some of Kashani's most impressive works, where he calculated pi to high accuracy and considered cubic equations, were produced during this time at the Samarkand observatory before his death in 1429.

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Kamran Moghbel
Professor: MohamadReza Riasati
Math 102
28, Jun, 2017
Ghyath al-Din Jamshid Kashani
Ghyath al-Din Jamshid Kashani was born about 1380 CE in
Kashan, Iran and died on 22 June 1429 in Samarkand,
Transoxania (now Uzbekistan). 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,
Ghyath al-Din Jamshid Kashani

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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

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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.
Letters which Kashani wrote in Persian to his father, who lived in Kashan, have survived. These
were written from Samarkand and give a wonderful description of the scientific life there. 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:

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... 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.
The last work by Kashani was The Treatise on the Chord and Sine which may have been
unfinished at the time of his death and then completed by Qazi Zadeh. In this work Kashani
computed sin 1 to the same accuracy as he had computed p in his earlier work. He also

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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....
Works Cited
Iranian Personalities: Ghyath Al-Din Jamshid Kashani. N.p., n.d. Web. 28 June 2017.