**Jamshid al-Kashi** was an Islamic mathematician who published some important teaching works and anticipated Stevin's work on decimals.

- Al-Kashi was born in Kashan which lies in a desert at the eastern foot of the Central Iranian Range.
- At the time that al-Kashi was growing up Timur (often known as Tamburlaine) was conquering large regions.
- al-Kashi lived in poverty, like so many others at this time, and devoted himself to astronomy and mathematics while moving from town to town.
- With the changing atmosphere, al-Kashi's life also improved markedly.
- The first event in al-Kashi'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.
- It is reasonable to assume that al-Kashi remained in Kashan where he worked on astronomical texts.
- Al-Kashi played this card to his advantage and brought himself into favour in the new era where patronage of the arts and sciences became popular.
- It was to Ulugh Beg that Al-Kashi dedicated his important book of astronomical tables Khaqani Zij which was based on the tables of Nasir al-Tusi.
- In the introduction al-Kashi says that without the support of Ulugh Beg he could not have been able to complete it.
- Al-Kashi also gives the tables of the longitudinal and latitudinal parallaxes for certain geographical latitudes, tables of eclipses, and tables of the visibility of the moon.
- Al-Kashi 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 Al-Kashi to join him at this school of learning in Samarkand, as well as around sixty other scientists including Qadi Zada.
- There is little doubt that al-Kashi 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 al-Kashi are undated they were written at a time when construction of the observatory had begun.
- In the letters al-Kashi praises the mathematical abilities of Ulugh Beg but of the other scientists in Samarkand, only Qadi Zada earned his respect.
- Usually these problems were too difficult for all except al-Kashi and Qadi Zada and on a couple of occasions only al-Kashi succeeded.
- It is clear that al-Kashi was the best scientist and closest collaborator of Ulugh Beg at Samarkand and, despite al-Kashi's ignorance of the correct court behaviour and lack of polished manners, he was highly respected by Ulugh Beg.
- Although al-Kashi had done some fine work before joining Ulugh Beg at Samarkand, his best work was done while in that city.
- It would be almost 200 years before van Ceulen surpassed Al-Kashi's accuracy with 20 decimal places.
- Al-Kashi'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 al-Kashi tries to give the necessary mathematics for those studying astronomy, surveying, architecture, accounting and trading.
- Al-Kashi uses decimal fractions in calculating the total surface area of types of muqarnas.
- Al-Kashi finds good methods to approximate the surface area and the volume of the shell forming the dome of the qubba.
- We mentioned above al-Kashi's use of decimal fractions and it is through his use of these that he has attained considerable fame.
- However, to claim that al-Kashi is the inventor of decimal fractions, as was done by many mathematicians following the work of Luckey, would be far from the truth since the idea had been present in the work of several mathematicians of al-Karaji's school, in particular al-Samawal.
- Al-Kashi can no longer be considered as the inventor of decimal fractions; it remains nonetheless, that in his exposition the mathematician, far from being a simple compiler, went one step beyond al-Samawal and represents an important dimension in the history of decimal fractions.
- There are other major results in the work of al-Kashi which were pointed out by Luckey.
- He found that al-Kashi had an algorithm for calculating nnnth roots which was a special case of the methods given many centuries later by Ruffini and Horner.
- The last work by al-Kashi was The Treatise on the Chord and Sine which may have been unfinished at the time of his death and then completed by Qadi Zada.
- In this work al-Kashi computed sin 1° to the same accuracy as he had computed π in his earlier work.
- But all these discoveries of al-Kashi's were long unknown in Europe and were studied only in the nineteenth and twentieth centuries by ...
- Let us end with one final comment on the al-Kashi's work in astronomy.
- We mentioned earlier the astronomical tables Khaqani Zij produced by al-Kashi.
- It is worth noting that Ulugh Beg also produced astronomical tables and sine tables, and it is almost certain that these tables were based on al-Kashi's tables and almost certainly produced with al-Kashi's help.

Born about 1380, Kashan, Iran. Died 22 June 1429, Samarkand, Transoxania (now Uzbek).

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Ancient Arab, Astronomy, Origin Iran, Number Theory, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive