 Kāshī, al

▪ Muslim astronomer and mathematicianIntroductionor alKāshānī, in full, Ghiyāth alDīn Jamshīd Mas'ūd alKāshīborn c. 1380, Kāshān, Persiadied June 22, 1429, Samarkand, Uzbekistanranks among the greatest mathematicians and astronomers in the Islamic world.Early lifeThe first event known with certainty in alKāshī's life is his observation of a lunar eclipse on June 2, 1406, from Kāshān. His earliest surviving work is Sullam alsamāʾ (1407; “The Stairway of Heaven”), an astronomical treatise dedicated to a local vizier. He dedicated the Mukhtaṣar dar ʿilmi hayʾat (1410–11; “Compendium of the Science of Astronomy”) to Iskander (executed in 1414), the sultan of Eṣfahan (Eṣfahān) and Fārs (both now located in Iran) and a member of the Timurid Dynasty. About 1413–14 alKāshī finished the Khāqānī Zīj. The first of his major works, this set of astronomical tables (zīj) was dedicated to Ulūgh Beg, the Khāqānī (“Supreme Ruler”) of Samarkand and grandson of the founder of the Timurid dynasty, the great Islamic leader Timur (1336–1405). Still seeking a patron, alKāshī completed two works in 1416, Risāla dar sharḥi ālāti raṣd (“Treatise on the Explanation of Observational Instruments”) and Nuzha alḥadāiq fī kayfiyya ṣanʾa alāla almusammā bi ṭabaq almanāṭiq (“The Garden Excursion, on the Method of Construction of the Instrument Called Plate of Heavens”), which describes a device (now known as an equatorium) that he invented for determining planetary positions. AlKāshī worked for some time in Herāt (now in Afghanistan) before finally receiving an invitation from Ulūgh Beg to come to Samarkand.Life in SamarkandFrom 1417 to 1420 Ulūgh Beg founded a madrasah (Islamic school for the study of theology, law, logic, mathematics, and natural science) in Samarkand to which he invited the greatest scholars of his realm. Following his arrival in about 1420, there can be no doubt that alKāshī was the leading astronomer and mathematician at the new institution. (Until the assasination of Ulūgh Beg in 1449, and the subsequent political repression, Samarkand was the most important centre of science in the Islamic realm.) In 1424 Ulūgh Beg, who was also an astronomer, began the construction of a great observatory at Samarkand, provisioned with the best equipment available. AlKāshī gives a vivid account of scholarly life at Samarkand during construction of the observatory in two undated letters to his father in Kāshān. In addition to including interesting information on the construction of the observatory building and the astronomical instruments, these letters characterize alKāshī as the closest collaborator and consultant of Ulūgh Beg.AlKāshī produced his greatest mathematical works after his arrival in Samarkand. In 1424 he completed the Risāla almuḥīṭīyya (“Treatise on the Circumference”), a computational masterpiece in which he determined the value of 2π to 9 sexagesimal places. (AlKāshī worked exclusively in base 60; his result is equivalent to 16 decimal places of accuracy, far eclipsing the 6 decimal places achieved by the Chinese mathematician Tsu Ch'ungchih [AD 430–501] and setting a record that lasted for almost 200 years.) In the introduction alKāshī observes that a small error in the estimated value of π results in a large error when calculating the circumference of enormous circles, such as the celestial sphere. In order to calculate the size of the universe with an error smaller than the width of a horse's hair (a standard Persian unit of measurement ^{1}/_{36} inch), alKāshī used a polygon with 3 × 2^{28} sides to estimate π.AlKāshī's bestknown work is the Miftāḥ alḥisāb (“Key of Arithmetic”), completed in 1427 and also dedicated to Ulūgh Beg. This encyclopedic work instructs in the solution of a wide range of problems from astronomy, surveying, and finance through the use of arithmetic—defined by alKāshī as “the science consisting of basic rules to find numerical unknowns from relevant known quantities.” The pedagogical excellence of the Miftāḥ alḥisāb is attested by the numerous copies made of it over the following centuries.In his third masterpiece, Risāla alwatar waʾljaib (“Treatise on the Chord and Sine”), he calculates the sine of 1° correct to 10 sexagesimal places. This precision was essential for the accuracy of Ulūgh Beg's Astronomical Tables. It is unclear, however, whether alKāshī completed the treatise himself or whether it was completed after his death by his colleague Qādī Zāde arRūmī (c. 1364–1436). AlKāshī was murdered outside the Samarkand observatory on June 22, 1429, probably on the command of Ulūgh Beg.Yvonne DoldSamploniusAdditional ReadingMohammad Bagheri, “A Newly Found Letter of alKāshī on Scientific Life in Samarkand,” Historia Mathematica, 24(3):241–256 (August 1997), is a translation of alKāshī's first letter to his father from the court of Ulūgh Beg.A translation of alKāshī's second letter to his father appears in E.S. Kennedy, “A Letter of Jamshīd alKāshī to His Father: Scientific Research and Personalities at a Fifteenth Century Court,” in E.S. Kennedy et al., Studies in the Islamic Exact Sciences (1983), pp. 722–744.Information on alKāshī's astronomical work can be found in E.S. Kennedy (trans.), The Planetary Equatorium of Jamshīd Ghiyāth alDīn alKāshī (1960); and “On the Contents and Significance of the Khāqānī Zīj by Jamshīd alDīn alKāshī,” Islamic Mathematics and Astronomy, vol. 84 (1998).Description of a Persian zīj can be found in E.S. Kennedy, “Spherical Astronomy in Kāshī's Khāqānī Zīj,” Zeitschrift für Geschichte der ArabischIslamischen Wissenschaften, 2:1–46 (1985).AlKāshī's mathematical achievements as well as his planetary equatorium are explained in J.L. Berggren, Episodes in the Mathematics of Medieval Islam (1986).Yvonne DoldSamplonius, “Practical Arabic Mathematics: Measuring the Muqarnas by alKāshī,” Centaurus 35(3–4):193–242 (1992), includes an English translation of a chapter from the Key of Arithmetic.Yvonne DoldSamplonius
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