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# Ghiyath al Din Jamshid Masud Kashi

(about 1380 in Kashan, Iran-22 June 1429 in Samarkand, Transoxania [now Uzbek])

Details of Jamshid Kashi's life and works are better known than many others from this period although details of his life are sketchy. One of the reasons we is that he dated many of his works with the exact date on which they were completed; another reason is that a number of letters which he wrote to his father have survived and give fascinating information.

Kashi was born in Kashan which lies in a desert at the eastern foot of theCentral Iranian Range. At the time that Kashi 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 inPersia with the capture of Herat. Timur died in 1405 and his empire was divided between his two sons, one of whom was Shahrokh.

While Timur was undertaking his military campaigns, conditions were very difficult with widespread poverty. Kashi 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 Shahrokh 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, Kashi's life also improved markedly. The first event in 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 Kashi remained in Kashan where he worked on astronomical texts. He was certainly in his home town on1 March 1407 when he completedSullam Al-sama' the text of which has survived. The full title of the work meansThe 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. 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. HisCompendium 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 Shahrokh made his own 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 Kashi dedicated his important book of astronomical tablesKhaqani Zij which was based on the tables of Nasir al-Tusi. In the introduction Kashi 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. See [J. Hamadanizadeh, The trigonometric tables of Kashi in his 'Zij-i Khaqani', Historia Math. 7 (1) (1980), 38-45] for a detailed discussion of this work.

TheKhaqani Zij also contains [Biography in Dictionary of Scientific Biography (New York 1970-1990)]:... detailed tables of the longitudinal motion of the sun, the moon, and the planets. 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.

Kashi had certainly found the right patron in Ulugh Beg since he founded a university for the study of theology and science atSamarkand in about 1420 and he sought out the best scientists to help with his project. Ulugh Beg invited 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 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.

Letters which Kashi wrote in Persian to his father, who lived in Kashan, have survived. These were written fromSamarkand and give a wonderful description of the scientific life there. In 1424 Ulugh Beg began the construction of an observatory inSamarkand and, although the letters by Kashi 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, see [M. Bagheri, A newly found letter of Kashi on scientific life in Samarkand,Historia Math. 24 (3) (1997), 241-256].

In the letters Kashi praises the mathematical abilities of Ulugh Beg but of the other scientists inSamarkand, only Qadi Zada earned his respect. Ulugh Beg led scientific meetings where problems in astronomy were freely discussed. Usually these problems were too difficult for all except Kashi and Qadi Zada and on a couple of occasions only Kashi succeeded. It is clear that al-Kashi was the best scientist and closest collaborator of Ulugh Beg atSamarkand and, despite Kashi's ignorance of the correct court behaviour and lack of polished manners; he was highly respected by Ulugh Beg. After Al-Kashi's death, Ulugh Beg described him as (see for example [Biography inDictionary of Scientific Biography (New York 1970-1990)]):... 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 Kashi had done some fine work before joining Ulugh Beg atSamarkand, 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 2 p 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 Kashi's accuracy with 20 decimal places.

Kashi's most impressive mathematical work was, however,The Key to Arithmetic which he completed on2 March 1427. The work is a major text intended to be used in teaching students in Samarkand, in particular Kashi tries to give the necessary mathematics for those studying astronomy, surveying, architecture, accounting and trading. The authors of [Biography inDictionary of Scientific Biography] 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.

Dold-Samplonius has discussed several aspects of Kashi'sKey to Arithmetic in [Y. Dold-Samplonius, The 15th century Timurid mathematician Ghiyath al-Din Jamshid al-Kashi and his computation of the Qubba, in S S Demidov et al. (eds),Amphora : Festschrift for Hans Wussing on the occasion of his 65th birthday (Basel- Boston- Berlin, 1992), 171-181], [Y. Dold-Samplonius, Practical Arabic mathematics : measuring the muqarnas by al-Kashi,Centaurus 35 (3-4) (1992), 193-242], and [Y. Dold-Samplonius, al-Kashi's measurement of Muqarnas, in Deuxième Colloque Maghrebin sur l'Histoire des Mathématiques Arabes (Tunis, 1990), 74-84]. (See also [Y. Dold-Samplonius,Qubba for al-Kashi : a videocassette (Providence, RI, 1995)]). For example the measurement of the muqarnas refers to a type of decoration used to hide the edges and joints in buildings such as mosques and palaces. The decoration resembles a stalactite and consists of three-dimensional polygons, some with plane surfaces, and some with curved surfaces. Kashi uses calculates decimal fractions in calculating the total surface area of types of muqarnas. The qubba is the dome of a funerary monument for a famous person. Kashi finds good methods to approximate the surface area and the volume of the shell forming the dome of the qubba.

We mentioned above Kashi's use of decimal fractions and it is through his use of these that he has attained considerable fame. The generally held view that Stevin had been the first to introduce decimal fractions was shown to be false in 1948 when P Luckey (see [P. Luckey, Die Rechnenkunst bei Gamsid b. Masud al-Kasi (Wiesbaden, 1951)]) showed that in theKey to Arithmetic Kashi gives as clear a description of decimal fractions asStevin does. However, to claim that 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.

Rashed (see [R. Rashed,The development of Arabic mathematics: between arithmetic and algebra (London, 1994)] or [R. Rashed,Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984)]) puts Kashi's important contribution into perspective. He shows that the main advances brought in by al-Kashi are:-

(1)The analogy between both systems of fractions; the sexagesimal and the decimal systems.

(2)The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such asp.

Rashed also writes (see [R. Rashed,The development of Arabic mathematics: between arithmetic and algebra (London, 1994)] or [R. Rashed,Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984)]):... 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 beyondal-Samawal and represents an important dimension in the history of decimal fractions.

There are other major results in the work of Kashi which were pointed out by Luckey. He found that Kashi had an algorithm for calculating nth roots which was a special case of the methods given many centuries later by Ruffini and Horner. In later work Rashed shows (see for example [R. Rashed,The development of Arabic mathematics: between arithmetic and algebra (London, 1994)] or [R. Rashed,Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984)]) that Al-Kashi was again describing methods which were present in the work of mathematicians of al-Karaji's school, in particular al-Samawal.

The last work by Kashi wasThe 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 Kashi computed sin 1 to the same accuracy as he had computed sin 1 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 Biruni had worked on it earlier. However, the iterative method proposed by Kashi was [Biography inDictionary of Scientific Biography]:... one of the best achievements in medieval algebra. ... But all these discoveries of Kashi's were long unknown in Europe and were studied only in the nineteenth and twentieth centuries by ... historians of science....

Let us end with one final comment on the Kashi's work in astronomy. We mentioned earlier the astronomical tablesKhaqani Zij produced by 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 Kashi's tables and almost certainly produced with Kashi's help.

### Article by:

J J O'Connor and E F Robertson**Taken from:**http://www-history.mcs.st-andrews.ac.uk/

**Also:**http://www.math.tamu.edu