- Jamshīd al-Kāshī
transl|ar|ALA|Ghiyāth al-Dīn Jamshīd ibn Masʾūd al-Kāshī (or transl|fa|ALA|Jamshīd Kāshānī,
Persian : غیاثالدین جمشید کاشانی) (c. 1380Kashan ,Iran –22 June 1429 Samarkand ,Transoxania ) was a Persian astronomer and mathematician.Biography
Al-Kashi was one of the best mathematicians in the
Islamic world . He was born in 1380, inKashan , which lies in a desert to the southeast of the Central Iranian range. This region was controlled by Tamurlane, better known as Timur, who was more interested in invading other areas than taking care of what he had. Due to this, al-Kashi lived in poverty during his childhood and the beginning years of his adulthood.The situation changed for the better when Timur died in 1405, and his son, Shah Rokh, ascended into power. Shah Rokh and his wife,
Goharshad , a Persian princess, were very interested in thesciences , and they encouraged their court to study the various fields in great depth. Their son,Ulugh Beg , was enthusiastic about science as well, and made some noted contributions in mathematics and astronomy himself. Consequently, the period of their power became one of many scholarly accomplishments. This was the perfect environment for al-Kashi to begin his career as one of the world’s greatest mathematicians.When he came into power, Ulugh Beg constructed the world’s most prestigious university at the time. Students from all over the
Middle East , and beyond, flocked to this academy inSamarkand , the capital of Ulugh Beg’s empire. Consequently, Ulugh Beg harvested many, many great mathematicians and scientists of theMuslim world . In 1414, al-Kashi took this opportunity to contribute vast amounts of knowledge to his people. His best work was done in the court of Ulugh Beg, and it is said that he was the king’s favourite student.Al-Kashi was still working on his book, called “Risala al-watar wa’l-jaib” meaning “The Treatise on the Chord and Sine”, when he died in 1429. Some scholars believe that Ulugh Beg may have ordered his murder, while others say he died a natural death. The details are rather unclear.
Astronomy
"Khaqani Zij"
Al-Kashi produced a "
Zij " entitled the "Khaqani Zij", which was based onNasir al-Din al-Tusi 's earlier "Zij-i Ilkhani ". In his "Khaqani Zij", al-Kashi thanks the Timurid sultan and mathematician-astronomerUlugh Beg , who invited al-Kashi to work at hisobservatory (seeIslamic astronomy ) and hisuniversity (seeMadrasah ) which taught Islamic theology as well asIslamic science . Al-Kashi produced sine tables to foursexagesimal digits (equivalent to eightdecimal places) of accuracy for each degree and includes differences for each minute. He also produced tables dealing with transformations betweencoordinate system s on thecelestial sphere , such as the transformation from theecliptic coordinate system to theequatorial coordinate system ."Treatise on Astronomical Observational Instruments"
In 1416,
al-Kashi wrote the "Treatise on Astronomical Observational Instruments", which described a variety of different instruments, including the triquetrum andarmillary sphere , the equinoctial armillary and solsticial armillary ofMo'ayyeduddin Urdi , thesine andversine instrument of Urdi, the sextant ofal-Khujandi , the Fakhri sextant at theSamarqand observatory, a double quadrantAzimuth -altitude instrument he invented, and a small armillary sphere incorporating analhidade which he invented. [Harv|Kennedy|1961|pp=104-107]Plate of Conjunctions
Al-Kashi invented the Plate of Conjunctions, an analog computing instrument used to determine the time of day at which
planetary conjunction s will occur, [Harv|Kennedy|1947|p=56] and for performinglinear interpolation .Planetary computer
Al-Kashi also invented a mechanical planetary computer which he called the Plate of Zones, which could graphically solve a number of planetary problems, including the prediction of the true positions in
longitude of theSun andMoon ,Harv|Kennedy|1950] and theplanet s in terms ofelliptical orbit s; [Harv|Kennedy|1952] thelatitude s of the Sun, Moon, and planets; and theecliptic of the Sun. The instrument also incorporated analhidade andruler . [Harv|Kennedy|1951]Mathematics
Law of cosines
In French, the
law of cosines is named "Théorème d'Al-Kashi" (Theorem of Al-Kashi), as al-Kashi was the first to provide an explicit statement of the law of cosines in a form suitable fortriangulation ."The Treatise on the Chord and Sine"
In "The Treatise on the Chord and Sine", al-Kashi computed sin 1° to nearly as much accuracy as his value for π, which was the most accurate approximation of sin 1° in his time and was not surpassed until
Taqi al-Din in the 16th century. Inalgebra andnumerical analysis , he developed aniterative method for solvingcubic equation s, which was not discovered in Europe until centuries later.A method algebraically equivalent to
Newton's method was known to his predecessorSharaf al-Dīn al-Tūsī . Al-Kāshī improved on this by using a form of Newton's method to solve to find roots of "N". Inwestern Europe , a similar method was later described by Henry Biggs in his "Trigonometria Britannica", published in 1633. [citation|title=Historical Development of the Newton-Raphson Method|first=Tjalling J.|last=Ypma|journal=SIAM Review|volume=37|issue=4|date=December 1995|publisher=Society for Industrial and Applied Mathematics|pages=531-551 [539] ]In order to determine sin 1°, al-Kashi discovered the following formula often attributed to
François Viète in the 16th century: [citation|title=Sherlock Holmes in Babylon and Other Tales of Mathematical History|last=Marlow Anderson, Victor J. Katz|first=Robin J. Wilson|publisher=Mathematical Association of America |year=2004|isbn=0883855461|page=139]"The Key to Arithmetic"
Computation of π
In one of his
numerical approximations of π , he correctly computed 2π to 9sexagesimal digits. ["Al-Kashi", author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256] This approximation of π is equivalent to 16decimal places of accuracy. This was far more accurate than the estimates earlier given inGreek mathematics (3 decimal places byArchimedes ),Chinese mathematics (7 decimal places byZu Chongzhi ) orIndian mathematics (11 decimal places byMadhava of Sangamagrama ). The accuracy of al-Kashi's estimate was not surpassed untilLudolph van Ceulen computed 20 decimal places of π nearly 200 years later.MacTutor|id=Al-Kashi|title=Ghiyath al-Din Jamshid Mas'ud al-Kashi]Decimal fractions
In discussing
decimal fractions , Struik states that (p. 7): [D.J. Struik, "A Source Book in Mathematics 1200-1800" (Princeton University Press, New Jersey, 1986). ISBN 0-691-02397-2]"The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphelet "De Thiende", published at Leyden in 1585, together with a French translation, "La Disme", by the Flemish mathematician
Simon Stevin (1548-1620), then settled in the NorthernNetherlands . It is true that decimal fractions were used by the Chinese many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal andsexagesimal fractions with great ease in his "Key to arithmetic" (Samarkand, early fifteenth century). [P. Luckey, "Die Rechenkunst bei Ğamšīd b. Mas'ūd al-Kāšī" (Steiner, Wiesbaden, 1951).] "Khayyam's triangle
In considering
Pascal's triangle , known in Persia as "Khayyam's triangle" (named afterOmar Khayyám ), Struik notes that (p. 21): [D.J. Struik, "op. cit."]"The Pascal triangle appears for the first time (so far as we know at present) in a book of 1261 written by
Yang Hui , one of the mathematicians of theSung dynasty inChina . [J. Needham, "Science and civilisation in China", III (Cambridge University Press, New York, 1959), 135.] The properties ofbinomial coefficient s were discussed by the Persian mathematician Jamshid Al-Kāshī in his "Key to arithmetic" of c. 1425. [Russian translation by B.A. Rozenfel'd (Gos. Izdat, Moscow, 1956); see also Selection I.3, footnote 1.] Both in China and Persia the knowledge of these properties may be much older. This knowledge was shared by some of theRenaissance mathematicians, and we see Pascal's triangle on the title page ofPeter Apian 's German arithmetic of 1527. After this we find the triangle and the properties of binomial coefficients in several other authors. [Smith, "History of mathematics", II, 508-512. See also our Selection II.9 (Girard).] "Notes
ee also
*
History of numerical approximations of π References
*Harvard reference
last=Kennedy
first=Edward S.
year=1947
title=Al-Kashi's Plate of Conjunctions
journal=Isis
volume=38
issue=1-2
pages=56-59
*Harvard reference
last=Kennedy
first=Edward S.
year=1950
title=A Fifteenth-Century Planetary Computer: al-Kashi's "Tabaq al-Manateq" I. Motion of the Sun and Moon in Longitude
journal=Isis
volume=41
issue=2
pages=180-183
*Harvard reference
last=Kennedy
first=Edward S.
year=1951
title=An Islamic Computer for Planetary Latitudes
journal=Journal of the American Oriental Society
volume=71
issue=1
pages=13-21
*Harvard reference
last=Kennedy
first=Edward S.
year=1952
title=A Fifteenth-Century Planetary Computer: al-Kashi's "Tabaq al-Maneteq" II: Longitudes, Distances, and Equations of the Planets
journal=Isis
volume=43
issue=1
pages=42-50
*MacTutor|id=Al-Kashi|title=Ghiyath al-Din Jamshid Mas'ud al-KashiExternal links
* [http://www.math-cs.cmsu.edu/~mjms/2000.2/azar5.ps A summary of "Miftah al-Hisab"]
* [http://www.iranchamber.com/personalities/jkashani/jamshid_kashani.php About Jamshid Kashani]
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