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| {{Infobox Muslim scholar |
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| <!-- Astronomer Category -->
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| notability = [[Persian peoples|Persian]] [[Muslim]] [[scholar]]|
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| era = [[Islamic Golden Age]]|
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| image =|
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| caption = |
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| name = '''Ghiyāth al-Dīn Jamshīd Kāshānī''' |
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| title= al-Kashi|
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| birth_date = 1380|
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| death_date = 22 June 1429 |
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| ethnicity = [[Persian people|Persian]]|
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| region = [[Iran]]|
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| Maddhab = |
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| school_tradition|= |
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| main_interests = [[Islamic astronomy|Astronomy]], [[Islamic mathematics|Mathematics]]|
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| notable_ideas= Pi decimal determination to the 16th place|
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| works = [[Sullam al-Sama]]|
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| influences =|
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| influenced =
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| }}
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| '''Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī''' (or '''al-Kāshānī''')<ref>[[A. P. Youschkevitch]] and [[B. A. Rosenfeld]]. "[http://www.encyclopedia.com/doc/1G2-2830902260.html al-Kāshī (al-Kāshānī), Ghiyāth al-Dīn Jamshīd Masʿūd]" ''[[Dictionary of Scientific Biography]]''.</ref> ({{lang-fa|غیاثالدین جمشید کاشانی}} ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 [[Kashan]], [[Iran]] – 22 June 1429 [[Samarkand]], [[Transoxania]]) was a [[Persian people|Persian]] [[Islamic astronomy|astronomer]] and [[Islamic mathematics|mathematician]].
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| ==Biography==
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| Al-Kashi was one of the best mathematicians in the [[Islamic world]]. He was born in 1380, in [[Kashan]], in central Iran. This region was controlled by [[Timur|Tamurlane]], better known as Timur. Al-Kashi lived in poverty during his childhood and the beginning years of his adulthood.
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| The situation changed for the better when Timur died in 1405, and his son, [[Shah Rukh (Timurid dynasty)|Shah Rokh]], ascended into power. Shah Rokh and his wife, [[Goharshad]], a Persian princess, were very interested in the [[sciences]], and they encouraged their court to study the various fields in great depth. 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.
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| Eight years after he came into power in 1409, their son, Ulugh Beg, founded an institute in [[Samarkand]] which soon became a prominent university. Students from all over the [[Middle East]], and beyond, flocked to this academy in the capital city of Ulugh Beg’s empire. Consequently, Ulugh Beg harvested many great mathematicians and scientists of the [[Muslim 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.
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| 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 unclear.
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| ==Astronomy==
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| ===''Khaqani Zij''===
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| Al-Kashi produced a ''[[Zij]]'' entitled the ''Khaqani Zij'', which was based on [[Nasir al-Din al-Tusi]]'s earlier ''[[Zij-i Ilkhani]]''. In his ''Khaqani Zij'', al-Kashi thanks the [[Timurid dynasty|Timurid]] sultan and mathematician-astronomer [[Ulugh Beg]], who invited al-Kashi to work at his [[observatory]] (see [[Islamic astronomy]]) and his [[university]] (see [[Madrasah]]) which taught [[Kalam|Islamic theology]] as well as [[Islamic science]]. Al-Kashi produced [[Trigonometric functions|sine]] tables to four [[sexagesimal]] digits (equivalent to eight [[decimal]] places) of accuracy for each degree and includes differences for each minute. He also produced tables dealing with transformations between [[coordinate system]]s on the [[celestial sphere]], such as the transformation from the [[ecliptic coordinate system]] to the [[equatorial coordinate system]].<ref name=MacTutor/>
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| ===''Astronomical Treatise on the size and distance of heavenly bodies''===
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| He wrote the book [[Sullam al-Sama]] on the resolution of difficulties met by predecessors in the determination of distances and sizes of [[Astronomical object|heavenly bodies]] such as the [[Earth]], the [[Moon]], the [[Sun]] and the [[Stars]].
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| ===''Treatise on Astronomical Observational Instruments''===
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| In 1416, [[al-Kashi]] wrote the ''Treatise on Astronomical Observational Instruments'', which described a variety of different instruments, including the [[Triquetrum (astronomy)|triquetrum]] and [[armillary sphere]], the [[Equinox|equinoctial]] armillary and [[Solstice|solsticial]] armillary of [[Mo'ayyeduddin Urdi]], the [[sine]] and [[versine]] instrument of Urdi, the [[Sextant (astronomical)|sextant]] of [[al-Khujandi]], the Fakhri sextant at the [[Samarqand]] observatory, a double quadrant [[Azimuth]]-[[altitude]] instrument he invented, and a small armillary sphere incorporating an [[alhidade]] which he invented.<ref>{{Harv|Kennedy|1961|pp=104–107}}</ref>
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| ====Plate of Conjunctions====
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| Al-Kashi invented the Plate of Conjunctions, an [[Analog computer|analog computing]] instrument used to determine the time of day at which [[planetary conjunction]]s will occur,<ref>{{Harv|Kennedy|1947|p=56}}</ref> and for performing [[linear interpolation]].<ref name=Kennedy/>
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| ====Planetary computer====
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| Al-Kashi also invented a mechanical planetary [[Analog computer|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 the [[Sun]] and [[Moon]],<ref name=Kennedy>{{Harv|Kennedy|1950}}</ref> and the [[planet]]s in terms of [[elliptical orbit]]s;<ref>{{Harv|Kennedy|1952}}</ref> the [[latitude]]s of the Sun, Moon, and planets; and the [[ecliptic]] of the Sun. The instrument also incorporated an [[alhidade]] and [[ruler]].<ref>{{Harv|Kennedy|1951}}</ref>
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| ==Mathematics==
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| ===Law of cosines===
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| In [[French language|French]], the [[law of cosines]] is named ''[[:fr:Théorème d'Al-Kashi|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 for [[triangulation]].
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| ===''The Treatise on the Chord and Sine''===
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| 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 Muhammad ibn Ma'ruf|Taqi al-Din]] in the 16th century. In [[algebra]] and [[numerical analysis]], he developed an [[iterative method]] for solving [[cubic equation]]s, which was not discovered in Europe until centuries later.<ref name=MacTutor/>
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| A method algebraically equivalent to [[Newton's method]] was known to his predecessor [[Sharaf al-Dīn al-Tūsī]]. Al-Kāshī improved on this by using a form of Newton's method to solve <math>x^P - N = 0</math> to find roots of ''N''. In [[western Europe]], a similar method was later described by Henry Biggs in his ''Trigonometria Britannica'', published in 1633.<ref>{{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]|doi=10.1137/1037125}}</ref>
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| In order to determine sin 1°, al-Kashi discovered the following formula often attributed to [[François Viète]] in the 16th century:<ref>{{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=0-88385-546-1|page=139}}</ref>
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| <math>\sin 3 \phi = 3 \sin \phi - 4 \sin^3 \phi\,\!</math>
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| ===''The Key to Arithmetic''===
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| ====Computation of 2π====
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| In his [[numerical approximations of π|numerical approximation]], he correctly computed 2π (or [[Tau (2π)|<math>\tau</math>]]) to 9 [[sexagesimal]] digits<ref>''Al-Kashi'', author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256</ref> in 1424,<ref name=MacTutor/> and he converted this approximation of 2π to 17 [[decimal]] places of accuracy.<ref>The statement that a quantity is calculated to <math>\scriptstyle n</math> sexagesimal digits implies that the maximal ''inaccuracy'' in the calculated value is less than <math>\scriptstyle 59/60^{n+1} + 59/60^{n+2} + \dots = 1/60^n</math> in the [[decimal]] system. With <math>\scriptstyle n=9</math>, Al-Kashi has thus calculated <math>\scriptstyle 2\pi</math> with a maximal ''error'' less than <math>\scriptstyle 1/60^{9}\approx 9.92\times 10^{-17} < 10^{-16}\,</math>. That is to say, Al-Kashi has calculated <math>\scriptstyle 2\pi</math> exactly up to and including the 16th place after the [[decimal separator]]. For <math>\scriptstyle 2\pi</math> expressed exactly up to and including the 18th place after the decimal separator one has: <math>\scriptstyle 6.283\,185\,307\,179\,586\,476</math>.</ref> This was far more accurate than the estimates earlier given in [[Greek mathematics]] (3 decimal places by [[Archimedes]]), [[Chinese mathematics]] (7 decimal places by [[Zu Chongzhi]]) or [[Indian mathematics]] (11 decimal places by [[Madhava of Sangamagrama]]). The accuracy of al-Kashi's estimate was not surpassed until [[Ludolph van Ceulen]] computed 20 decimal places of π nearly 200 years later.<ref name=MacTutor>{{MacTutor|id=Al-Kashi|title=Ghiyath al-Din Jamshid Mas'ud al-Kashi}}</ref> It should be noted that al-Kashi's goal was not to compute the circle constant with as many digits as possible but to compute it so precisely that the circumference of the largest possible circle (ecliptica) could be computed with highest desirable precision (the diameter of a hair).
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| ====Decimal fractions====
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| In discussing [[decimal fractions]], [[Dirk Jan Struik|Struik]] states that (p. 7):<ref name="Struik1986">D.J. Struik, ''A Source Book in Mathematics 1200-1800'' (Princeton University Press, New Jersey, 1986). ISBN 0-691-02397-2</ref>
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| <blockquote>
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| "The introduction of decimal fractions as a common computational practice can be dated back to the [[Flemish Region|Flemish]] pamphlet ''De Thiende'', published at [[Leiden|Leyden]] in 1585, together with a French translation, ''La Disme'', by the Flemish mathematician [[Simon Stevin]] (1548-1620), then settled in the Northern [[Netherlands]]. It is true that decimal fractions were used by the [[Chinese mathematics|Chinese]] many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal and [[sexagesimal]] fractions with great ease in his ''Key to arithmetic'' (Samarkand, early fifteenth century).<ref>P. Luckey, ''Die Rechenkunst bei Ğamšīd b. Mas'ūd al-Kāšī'' (Steiner, Wiesbaden, 1951).</ref>"
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| </blockquote>
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| ====Khayyam's triangle====
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| In considering [[Pascal's triangle]], known in Persia as "Khayyam's triangle" (named after [[Omar Khayyám]]), Struik notes that (p. 21):<ref name="Struik1986"/>
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| "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 the [[Sung dynasty]] in [[China]].<ref>J. Needham, ''Science and civilisation in China'', III (Cambridge University Press, New York, 1959), 135.</ref> The properties of [[binomial coefficient]]s were discussed by the Persian mathematician Jamshid Al-Kāshī in his ''Key to arithmetic'' of c. 1425.<ref>Russian translation by B.A. Rozenfel'd (Gos. Izdat, Moscow, 1956); see also Selection '''I.3''', footnote '''1'''.</ref> Both in China and Persia the knowledge of these properties may be much older. This knowledge was shared by some of the [[Renaissance]] mathematicians, and we see [[Blaise Pascal|Pascal]]'s triangle on the title page of [[Peter Apian]]'s [[German language|German]] arithmetic of 1527. After this we find the triangle and the properties of binomial coefficients in several other authors.<ref>Smith, ''History of mathematics'', II, 508-512. See also our Selection '''II.9''' (Girard).</ref>"
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| </blockquote>
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| == Biographical film ==
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| In 2009 [[IRIB]] produced and broadcast (through Channel 1 of IRIB) a biographical-historical film series on the life and times of Jamshid Al-Kāshi, with the title ''[[The Ladder of the Sky]]'' <ref>''The narrative by Latifi of the life of the celebrated Iranian astronomer in 'The Ladder of the Sky' '', in Persian, Āftāb, Sunday, 28 December 2008, [http://www.aftab.ir/news/2008/dec/28/c5c1230463130_art_culture_media_serial.php].</ref><ref>''IRIB to spice up Ramadan evenings with special series'', Tehran Times, 22 August 2009, [http://www.tehrantimes.com/index_View.asp?code=201568].</ref> (''Nardebām-e Āsmān'' <ref>The name ''Nardebām-e Āsmān'' coincides with the [[Persian language|Persian]] translation of the title ''Soll'am-os-Samā' '' (سُلّمُ السَماء) of a scientific work by Jamshid Kashani written in [[Arabic language|Arabic]]. In this work, which is also known as ''Resāleh-ye Kamālieh'' (رسالهٌ كماليه), Jamshid Kashani discusses such matters as the diameters of [[Earth]], the [[Sun]], the [[Moon]], and of the [[stars]], as well as the distances of these to Earth. He completed this work on 1 March 1407 CE in Kashan.</ref>). The series, which consists of 15 parts of each 45 minutes duration, is directed by Mohammad-Hossein Latifi and produced by Mohsen Ali-Akbari. In this production, the role of the adult Jamshid Al-Kāshi is played by Vahid Jalilvand.<ref>''The programmes of the Holy month of Ramadan, Channel 1'', in Persian, 19 August 2009, [http://ch1.iribtv.ir/index.php?option=com_content&task=view&id=5246&Itemid=265]. Here the name "Latifi" is incorrectly written as "Seifi".</ref><ref>''Dr Velāyati: 'The Ladder of the Sky' is faithful to history'', in Persian, Āftāb, Tuesday, 1 September 2009, [http://www.aftab.ir/news/2009/sep/01/c5c1251816526_art_culture_media_serial.php].</ref><ref>Fatemeh Udbashi, ''Latifi's narrative of the life of the renowned Persian astronomer in 'The Ladder of the Sky' '', in Persian, Mehr News Agency, 29 December 2008, [http://www.mehrnews.ir/NewsPrint.aspx?NewsID=808056].</ref>
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| ==Notes==
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| {{reflist|2}}
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| ==See also==
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| *[[History of numerical approximations of π]]
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| == References ==
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| *{{Citation
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| |last=Kennedy
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| |first=Edward S.
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| |year=1947
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| |title=Al-Kashi's Plate of Conjunctions
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| |journal=[[Isis (journal)|Isis]]
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| |volume=38
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| |issue=1–2
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| |pages=56–59
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| |doi=10.1086/348036
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| }}
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| *{{Citation
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| |last=Kennedy
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| |first=Edward S.
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| |year=1950
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| |title=A Fifteenth-Century Planetary Computer: al-Kashi's "Tabaq al-Manateq" I. Motion of the Sun and Moon in Longitude
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| |journal=[[Isis (journal)|Isis]]
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| |volume=41
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| |issue=2
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| |pages=180–183
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| |doi=10.1086/349146
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| }}
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| *{{Citation
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| |last=Kennedy
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| |first=Edward S.
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| |year=1951
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| |title=An Islamic Computer for Planetary Latitudes
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| |journal=[[Journal of the American Oriental Society]]
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| |volume=71
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| |issue=1
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| |pages=13–21
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| |doi=10.2307/595221
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| |publisher=American Oriental Society
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| |jstor=595221
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| }}
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| *{{Citation
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| |last=Kennedy
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| |first=Edward S.
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| |year=1952
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| |title=A Fifteenth-Century Planetary Computer: al-Kashi's "Tabaq al-Maneteq" II: Longitudes, Distances, and Equations of the Planets
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| |journal=[[Isis (journal)|Isis]]
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| |volume=43
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| |issue=1
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| |pages=42–50
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| |doi=10.1086/349363
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| }}
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| *{{MacTutor|id=Al-Kashi|title=Ghiyath al-Din Jamshid Mas'ud al-Kashi}}
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| ==External links==
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| * {{cite encyclopedia | editor = Thomas Hockey et al | last = Schmidl | first = Petra G. | title=Kāshī: Ghiyāth (al‐Milla wa‐) al‐Dīn Jamshīd ibn Masʿūd ibn Maḥmūd al‐Kāshī [al‐Kāshānī] | encyclopedia = The Biographical Encyclopedia of Astronomers | publisher = Springer | year = 2007 | location = New York | pages = 613–5 | url=http://islamsci.mcgill.ca/RASI/BEA/Kashi_BEA.htm | isbn=978-0-387-31022-0}} ([http://islamsci.mcgill.ca/RASI/BEA/Kashi_BEA.pdf PDF version])
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| *[http://www.math-cs.cmsu.edu/~mjms/2000.2/azar5.ps A summary of "Miftah al-Hisab"]
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| *[http://www.iranchamber.com/personalities/jkashani/jamshid_kashani.php About Jamshid Kashani]
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| {{Islamic astronomy}}
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| {{Islamic mathematics}}
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME = Kashi, Jamshid
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = Persian mathematician and astronomer
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| | DATE OF BIRTH = 1380
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| | PLACE OF BIRTH =
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| | DATE OF DEATH = 1429
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| | PLACE OF DEATH =
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| }}
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| {{DEFAULTSORT:Kashi, Jamshid}}
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| [[Category:1380 births]]
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| [[Category:1429 deaths]]
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| [[Category:People from Kashan]]
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| [[Category:Astronomers of medieval Islam]]
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| [[Category:Persian astronomers]]
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| [[Category:Mathematicians of medieval Islam]]
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| [[Category:14th-century mathematicians]]
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| [[Category:15th-century mathematicians]]
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| [[Category:Medieval Persian mathematicians]]
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| [[Category:Medieval Persian astrologers]]
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| [[Category:15th-century astronomers]]
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| [[Category:15th-century Iranian people]]
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